The first thing is to stop throwing Parts and time at it.
You need to have an established diagnostic procedure.. and preferably a logical one that works best.
Spraying the car cleaner into the carburetor intake was certainly the first thing to do after making sure the spark plug wire was actually on and when it doesn't run at least for a few seconds after doing that two or three times then you know the engine is not mechanically sound so it doesn't do any good to worry about the carburetor at this point or even consider putting a new one on or even clean the bowl and stuff like you did to start with.
Since most of the time when an engine doesn't run it's lack of fuel delivery on its own spraying carb cleaner into the intake carb hole below the air filter is the easiest way to determine if the engine is mechanically sound and it is a carburetor issue but you have determined that you're supposed more than likely is not.
Those particular engines have a common problem of the head gasket failing and leaking and I might be able to hear it puffing on yours but since it's from the rear I can't tell.
You can put your hand around the side of the front head where it meets the block and various locations and have someone pull it and you will hear and feel the air blowing out if it's a leaky head gasket.
Then you can pull the spark plug out and put your finger in the hole, do not get a compression tester and do not bother to go to the toolbox and screw one in because it's just a waste of time...lol...and with your finger stuck in the edge of the hole have someone pull the rope or pull it with your other hand which I'm really good at and see if it blows your finger out of the hole or see if the compression feels low.
If it has low to no compression it could also be a valve issue but it could also be the head gasket but normally you can hear that puffing out around the side.
Those are unlikely to have a flywheel key sheared but it is possible.
Normally when one of these engines on the time Master does not start and run with carb cleaner or dripping some fuel in there etc then it is either the head gasket, or something with the valves or the head like that push rods, rocker arm jumped off the push rod, loosen rocker arm studs, slid valve guide, or popped valve seat all of which would be considered a valve train / head issue.
Also, all of which can be fixed without buying any new parts!
Too many people jump on Amazon and pay $138 or more for a head when they have a problem like this when it's just not necessary.
And to answer your other question, no starting fluid is not worth a shot.
If it won't run on carb cleaner then it won't run on starting fluid and starting fluid is way hard on the engines when they do start and run on it. I do not own a can of starting fluid and I do thousands of mower repairs and diagnoses a year.
