Valve/tappet clearance will not affect timing or air/fuel passage enough to matter so long as the clearance is not zero. At the very worst, it could cause excessive valve tap.
The Briggs and Stratton shop manual says the clearance should be .005 to .007 for the intake and .007 to .009" for the exhaust valve. A couple thousandths of an inch difference won't amount to enough to make it run or not run.
The only reason I checked it at all was to make sure the clearance hadn't shrunk down to zero which could hold a valve open on the compression stroke. Valve clearances will vary with engine temp as the aluminum block expands as the engine warms up. Since aluminum expands at a different rate than steel, the valve clearances will change with temperature. This is normal.
What's not normal is it losing spark below full throttle. The coil for some reason seems to stop making spark below about 2000 rpm.
At first I thought maybe the 'fixed throttle' engine's carb maybe didn't have a full idle circuit but it runs completely normal and will idle on the other mower with a throttle control. The issue has to be spark to at least some degree, but I can't see how movement of the mower would affect that.
If anything was greatly amiss inside the engine I'd have noise. The cam timing isn't changing or moving, the crank doesn't have excess end play, and the valves are not sticking or held open by 'too tight' valve clearances, and its got good compression.
It makes me think that maybe it won't restart after its hot because the recoil can't get it spinning fast enough to make spark once its hot. But none of that explains why it'll run through a tank of fuel if I just let it sit there on the trailer till it runs out of gas. It never misses a beat. However, if I walk up to it, and push it forward and and back real fast it sputters and gets unstable, it starts acting like its running out of gas and never recovers, it will die even if I leave it sit again. After that point it will restart over and over running a shorter amount of time each time until it will only start and stall. Yet if I left it sit untouched it would have run till the tank was empty.
Like I said before, my first impression was that there was a puddle of water or something on the bottom of the tank, or in the float bowl being sucked up when the momentum forward stopped suddenly. But I've changed the carb, fuel tank, fuel, fuel line and tank several times. I've got a half dozen fuel tanks on the bench here, four carbs, 7 coils, four flywheels, and a box of spark plugs, all of which ran fine on other engines.
If there was something loose enough inside the engine where momentum would make it shut down, I'd hear it, and it likely wouldn't sputter and carry on for a minute before it died.
Someone local said check the float level but the floats in these are fixed, there's no adjustment. They likely don't use enough fuel to require or be affected by a minor difference in fuel level in the fuel bowl.
On the good mower, if I blip the throttle a bit on top of the carb it'll rev the engine higher and return to the fixed RPM.
If I do that on this mower, it will not rev, it dies with any throttle movement It bogs and stalls as the governor tries to return the throttle to the correct speed. On the good mower, I can easily over rev the engine and it returns to the set speed instantly. This mower revs ever so slightly then shuts off as I let go of the throttle.
Keep in mind that this problem came on slowly a couple of years ago when I stopped using it. It started acting up only a bit and got very sensitive to what fuel I used. It wouldn't run on older gas, it got hard to start on month old gas while the other one would burn anything I dumped in the tank. over the course of a few weeks it went from being finicky about gas to stalling every 20 minutes then finally it got to where its at now. That's when I put it away and used a different mower. It didn't happen all at once, as if something broke, it came on slow and got worse till it got to where its at now.
When it happened, I really just thought the carb needed cleaning. Which is what I did first a few weeks ago. That got it running again but since that nothing I've done has changed the situation.
Also, the point when it shuts off the first time isn't really long enough for it to get 'hot', weren't talking three to five passes across a 60ft wide lawn and around a few bushes when it dies the first time. The good mower will run all day tank after tank without any issues.