How about giving us the model number of the mower so we can see what setup you have. Also does the mower have an electric clutch, and have you checked it for seized bearings. I am wondering if the problem is not the engine but some other parasitic load causing issues.
About a dozen time a year some one comes in with a mower that will not crank.
It rotates till it gets to compression and stops,
Usually they have spent near $ 1000 replacing parts , some times replacing the replacements.
The solution is always the same
Either adjust the valves or replace the cam shaft.
Usually like you it has been happening for years and they have just been working around it.
So in your case you will either have a faulty lifter or a worn out cam.
FWIW to test weather you have a problem with the cranking you simply hook up a jumper from the + on the starter motor to the + on the battery
If the results are the same as turning the key, the cranking circuit is fine.
You double check the ground by putting a jumper from the battery - to a good grounding spot ( I like the oil drain plug ) and see if that makes a difference.
Kohlers tend to suffer from crud build up on the bottom of the starter more than other brands .
The simple test it to feel the heat after cranking
The power goes through the field coils so the side of the starter should be a lot hotter than the bottom.
However you said you replaced the starter so provided you bought a certified good one from a real shop and not the cheapest one on Ebay a bad starter could be ruled out.
Have you removed and cleaned where the negative cable bolts to the frame?
A cam shaft is not a difficult job, but a big one with all of the cleaning & adjusting and of course it is an engine out job.
That is a fairly old mower so it is up to you if you feel it is worth it.
No problems with B & S cause their parts are cheap but Kohler parts are much more expensive.
My customers generally spring for a replacement engine & I give them a reasonable trade in on the dead one.
I keep telling myself that I will pull few down & make a good one or two out of them but it rarely happens .
Mostly I sell 3 to a novice racer for what I traded them for.
Parts books & service books are available directly from Kohler as a free download but I find it easier to go here https://www.mymowerparts.com/pdf/Kohler-Service-and-Repair-Manuals/
To do a cam shaft you will also need at least a sump gasket, rocker gasket & crankshaft oil seal.
In theory you can leave the head on.
I always take it off ( customer pays for the new gasket in any case ) to check the bore as I give 12 months on repairs.
If you want to go through with it then take lots of photos, particularly of the controls & springs on the controls.
We will be here to walk you through it.
Not particularly difficult but a first timers should allocate a weekend or two as just getting the drive pulleys off can be an all day affair.
In the dark remove the battery ground terminal.
If the rectifier has an open circuit diode the alternator will try to motor when the engine is not running which flattens the battery.
Other than that we ned the test voltages every one has been asking you for.
In the dark remove the battery ground terminal.
If the rectifier has an open circuit diode the alternator will try to motor when the engine is not running which flattens the battery.
Other than that we ned the test voltages every one has been asking you for.
Appologies I was confusing you with another non starting Kohler engine
Voltage across the battery terminals
1) everything off
2) ignition on
3) ignition on with engine running full speed
4) engine running full speed blades on
Do the same between the purple wire and a grounding point on the engine
Do the same between the purple wire and the battery neg
Then AC Volts between each of the white or yellow or black wires coming into the regulator should be over 28V AC
Followed by AC Volts between each wire & ground. Should be 1/2 the above
Then AC between the purple wire & ground. Should be 0
The 2 white wires are the AC that is being generated by the armature and between them you should see a bit better than double the DC charge volts ~ 28 V AC.
The middle wire is Rectified DC output to the battery and yours looks good if you are getting 14,3 V engine running with blades engaged.
So we can cross the battery off the criminal suspects list.
No sparks when disconnecting crosses a phantom load off the suspects list
A 1 volt drop is starting to look like a bad battery.
Try cranking the engine by connecting the starter directly to the battery .
You will get a big flash when you make the connection, so hook it to the starter first & then touch it to the battery.
If the engine cranks then you have a problem with the wiring in the mower.
Mowers left outside get corrosion happening in the heavy battery cables, inside the insulation because water wicks down between the wires & between the wires & the insulation.
These will test good with a volt meter but fail to carry the 100 amps needed to crank the engine.
Purple wire is the black wire in the middle of the rectifier.
Not seen a black one before but there are a lot of things I have not seen before which is why I am here.
Take the spark plug out & try to crank the engine.
If it cranks plug out but won't crank plug in the you have confirmed an internal engine problem.
Sorry I should have suggested this a lot earlier but it is one of those things that happens when you are looking at a screen & not a mower.
Wouldn't start under own power but did hear solenoid click. Jumped off non-running car battery and started well