As a couple of people have said, you need to back up to the basic diagnosing when it does this.
It could be any of the things others have mentioned and it could have been the bad coil from the beginning like you thought of but normally a bad coil will go 12 to 15 minutes minimum before it breaks down. Anything shorter than that it's probably not a coil and they also typically won't restart even in 15 or 20 minutes but if they do the only run for maybe a minute or so much much shorter than the first time.
They usually have to cool off for hours or overnight before they repeat the process.
But at this point you need to get it to mess up again, which seems like that won't be a problem, and then do the same basic troubleshooting you should do anytime an engine doesn't run or dies.
Always start with fuel!
This is because well over 80% of the issues do boil down to fuel and most of those end up being not enough fuel so don't worry about spark first thing as it's usually a waste of time despite the fact that everybody's go to is to check for spark or replace the spark plug which is typically just a waste of a few dollars for a plug.. yes, I'm opinionated but I'm right. Lol customers call me every single day talking about they try to plug but.. you know the rest of the sentence so my theory is that replacing a new spark plug is usually just a waste of four or five bucks. Also it literally boils my blood on how many engines come in with damaged threads or cross threaded spark plugs only screwed in three threads because the people screwed it up and the process of trying to change it. Step away from the toolbox sir, step away from the toolbox.... Is normally what I'm thinking.
So, when yours messes up be ready with a spray bottle of gas or a can of carb cleaner to spray into the intake to see if it instantly fires back up if it doesn't when you crank it right then which I think is the case in it's normal failure mode.
If it fires right up and runs as long as you spray the external fuel source into it then you know it's not a mechanical engine problem or a coil ignition issue.
You'll probably find this is the case and now it takes more intense troubleshooting because fuel issues can be tricky and you've already done so much that should have fixed it but.. yet here we are.
This is another thing in this world, no matter how many things aren't supposed to happen, if you do it long enough you'll see things that just do..
The gas cap not being vented causes this from time to time as does debris in the outlet of the fuel tank where the fuel line presses on where grass and even ladybugs, insects, etc gets down in there and you can't see it.
Inner fuel line delamination and internal prolapse of the fuel lines also cause flow issues.. but probably more common than this would be the inlet seats swelling up but most riding lawn mowers don't have inlet seats that can swell up because that's typically a push mower thing with the solid metal needle and then the rubber inlet seed and most riders have a rubber tipped inlet needle and that rarely swells up.
But debris can get there at the inlet of the carburetor and I have seen it three times on a brand new machine where it would not run properly because of this and what came out was a very small packed-in piece of cellulose or cotton looking wadding I would estimate 1/4 the size of a small pea.
I have a picture somewhere of the last one I pulled out.
It could also be the carburetor solenoid which is a common problem too if you have one on the bottom of the carburetor, which most do.
These things get hot and break down or the ones with the rubber tip will have the rubber get hard and lose its Bond or stickiness cohesion to the metal and it will vibrate up on the actual shaft but it won't fall off.
I have had several of these occur like this which were very intermittent but they would repeatedly do this over and over.
Very rarely will a certain brand, even the cheapest fuel filter cause a problem..
There was a TSB for Kohler on the command I think it was just a singles but maybe both years ago about this and so a lot of people are still saying you have to use OEM filters but I think this is one of the least likely possibilities of the problem.
Some cheap filters though do you have two or three different sizes that are stair stepped as you push the fuel line in and that's kind of bad because most the time the smallest hole of the end is smaller than the OEM so it would probably be a good idea to cut that filter off further up to where it actually matches your fuel line but I still think most will flow enough with the smallest hole and that's probably the thought process. These mowers don't need that fast of a flow to keep the bowl full but once you hit the bare minimum even slightly below it will cause running problems.
Be very careful when you're blowing things out around your gas tank. Never have the cap on when you're blowing backwards through it because it will overpressurize the tank and you can split the seams right out of them!
I had to replace a fuel tank for a man one time on a craftsman that went underneath the seat and he purchased the tank himself at the tune of almost $100 from Craftsman because he blew air backwards through it.
Testing flow though is a good idea and it sounds like you have a gravity-fed system so you should have a good strong steady flow when you remove the fuel line from the carburetor.
If you do then the next place would be to take the bowl off and work the float with your fingertip to see if you have a good flow coming out of the inlet and while you're there you can inspect the solenoid if it has one.
Now if nothing shows up from here it's time to do a test by removing your factory gas tank and fuel line from the equation.
Rig up a small external test tank.
You can use any old gas tank even when you take off of a push mower.
Just make sure it's clean inside and blow it out good with compressed air.
Get you a couple of foot piece of fuel line etc and stick on that and to the carburetor and fill the tank fairly full so you'll have some gravity and pressure working to your benefit. Either hold it or rig away to wedge it by the seat or whatever or tape it or bungee cord it or rig you a rod that you can stick up in the air a little bit and tape it or zip tie it to and start it and run it off this auxiliary tank.
This shouldn't take over 10 or 15 minutes to set up and if you get perfect results from this then you can start looking towards your fuel tank and the fuel line getting to the carb.
I have personally witnessed the gas caps not venting causing them to stumble and shut off, I have also personally pulled three ladybugs out of the gas tank fuel line exit, and I have also pulled multiple over 3-in pieces of grass out of a Cub cadet zero turn that exhibited the same fuel starvation issues but randomly. I may or may not have witnessed a Kohler command on an older LT John Deere that had an aftermarket fuel filter that was causing problems.
The machine did mess up but I believe the fuel cap vent had more to do with it than the fuel filter but it did mess up and it did straighten out but I didn't use a scientific procedural method to determine exactly which one was the cause when I did them both at the same time or disturb the cap in any way while I changed the fuel filter.
Oh, and on fuel filters, they are almost never full of fuel!
So many people think there's a problem because they only see like a quarter to a half in there but that's what most of them look like and they run just fine.
It could be any of the things others have mentioned and it could have been the bad coil from the beginning like you thought of but normally a bad coil will go 12 to 15 minutes minimum before it breaks down. Anything shorter than that it's probably not a coil and they also typically won't restart even in 15 or 20 minutes but if they do the only run for maybe a minute or so much much shorter than the first time.
They usually have to cool off for hours or overnight before they repeat the process.
But at this point you need to get it to mess up again, which seems like that won't be a problem, and then do the same basic troubleshooting you should do anytime an engine doesn't run or dies.
Always start with fuel!
This is because well over 80% of the issues do boil down to fuel and most of those end up being not enough fuel so don't worry about spark first thing as it's usually a waste of time despite the fact that everybody's go to is to check for spark or replace the spark plug which is typically just a waste of a few dollars for a plug.. yes, I'm opinionated but I'm right. Lol customers call me every single day talking about they try to plug but.. you know the rest of the sentence so my theory is that replacing a new spark plug is usually just a waste of four or five bucks. Also it literally boils my blood on how many engines come in with damaged threads or cross threaded spark plugs only screwed in three threads because the people screwed it up and the process of trying to change it. Step away from the toolbox sir, step away from the toolbox.... Is normally what I'm thinking.
So, when yours messes up be ready with a spray bottle of gas or a can of carb cleaner to spray into the intake to see if it instantly fires back up if it doesn't when you crank it right then which I think is the case in it's normal failure mode.
If it fires right up and runs as long as you spray the external fuel source into it then you know it's not a mechanical engine problem or a coil ignition issue.
You'll probably find this is the case and now it takes more intense troubleshooting because fuel issues can be tricky and you've already done so much that should have fixed it but.. yet here we are.
This is another thing in this world, no matter how many things aren't supposed to happen, if you do it long enough you'll see things that just do..
The gas cap not being vented causes this from time to time as does debris in the outlet of the fuel tank where the fuel line presses on where grass and even ladybugs, insects, etc gets down in there and you can't see it.
Inner fuel line delamination and internal prolapse of the fuel lines also cause flow issues.. but probably more common than this would be the inlet seats swelling up but most riding lawn mowers don't have inlet seats that can swell up because that's typically a push mower thing with the solid metal needle and then the rubber inlet seed and most riders have a rubber tipped inlet needle and that rarely swells up.
But debris can get there at the inlet of the carburetor and I have seen it three times on a brand new machine where it would not run properly because of this and what came out was a very small packed-in piece of cellulose or cotton looking wadding I would estimate 1/4 the size of a small pea.
I have a picture somewhere of the last one I pulled out.
It could also be the carburetor solenoid which is a common problem too if you have one on the bottom of the carburetor, which most do.
These things get hot and break down or the ones with the rubber tip will have the rubber get hard and lose its Bond or stickiness cohesion to the metal and it will vibrate up on the actual shaft but it won't fall off.
I have had several of these occur like this which were very intermittent but they would repeatedly do this over and over.
Very rarely will a certain brand, even the cheapest fuel filter cause a problem..
There was a TSB for Kohler on the command I think it was just a singles but maybe both years ago about this and so a lot of people are still saying you have to use OEM filters but I think this is one of the least likely possibilities of the problem.
Some cheap filters though do you have two or three different sizes that are stair stepped as you push the fuel line in and that's kind of bad because most the time the smallest hole of the end is smaller than the OEM so it would probably be a good idea to cut that filter off further up to where it actually matches your fuel line but I still think most will flow enough with the smallest hole and that's probably the thought process. These mowers don't need that fast of a flow to keep the bowl full but once you hit the bare minimum even slightly below it will cause running problems.
Be very careful when you're blowing things out around your gas tank. Never have the cap on when you're blowing backwards through it because it will overpressurize the tank and you can split the seams right out of them!
I had to replace a fuel tank for a man one time on a craftsman that went underneath the seat and he purchased the tank himself at the tune of almost $100 from Craftsman because he blew air backwards through it.
Testing flow though is a good idea and it sounds like you have a gravity-fed system so you should have a good strong steady flow when you remove the fuel line from the carburetor.
If you do then the next place would be to take the bowl off and work the float with your fingertip to see if you have a good flow coming out of the inlet and while you're there you can inspect the solenoid if it has one.
Now if nothing shows up from here it's time to do a test by removing your factory gas tank and fuel line from the equation.
Rig up a small external test tank.
You can use any old gas tank even when you take off of a push mower.
Just make sure it's clean inside and blow it out good with compressed air.
Get you a couple of foot piece of fuel line etc and stick on that and to the carburetor and fill the tank fairly full so you'll have some gravity and pressure working to your benefit. Either hold it or rig away to wedge it by the seat or whatever or tape it or bungee cord it or rig you a rod that you can stick up in the air a little bit and tape it or zip tie it to and start it and run it off this auxiliary tank.
This shouldn't take over 10 or 15 minutes to set up and if you get perfect results from this then you can start looking towards your fuel tank and the fuel line getting to the carb.
I have personally witnessed the gas caps not venting causing them to stumble and shut off, I have also personally pulled three ladybugs out of the gas tank fuel line exit, and I have also pulled multiple over 3-in pieces of grass out of a Cub cadet zero turn that exhibited the same fuel starvation issues but randomly. I may or may not have witnessed a Kohler command on an older LT John Deere that had an aftermarket fuel filter that was causing problems.
The machine did mess up but I believe the fuel cap vent had more to do with it than the fuel filter but it did mess up and it did straighten out but I didn't use a scientific procedural method to determine exactly which one was the cause when I did them both at the same time or disturb the cap in any way while I changed the fuel filter.
Oh, and on fuel filters, they are almost never full of fuel!
So many people think there's a problem because they only see like a quarter to a half in there but that's what most of them look like and they run just fine.