One possible cause of your problem is varnish in the carburetor, hidden where it can't be seen. If you add some gum-out or similar additive to the gasoline, the carb should clean itself out, but not immediately. It can take one to four hours of operation for carb cleaner to successfully remove the varnish deposit.

Mechanically, the symptoms don't make sense.

Might be a plugged breather hole in the gas cap! Or, the little hole in the bolt that holds the carb bowl on, if so equipped!

MAIN METERING CLOGGED BY PUSHING PRIMER THAT PUSHES FUEL UP TO THE THROAT OF THE CARB. RICK

If it was a clogged jet, still unclogged, priming it all day would never make it run semi-normal on its own.

This sounds more like a choke issue, but there is no choke.

If it were any other fuel starvation issue, it wouldn't be coaxed to run by priming it.

So, I'm thinking of some air/fuel problem orleak that is almost overcome after the engine warms up or is able to maintain full speed on its own after warming up.

It's that stupid bulb primer!
I have the same problem, rough to get it started and then once it's warmed up, it's ok.

I tear it apart and can get one or two starts out of it before it clogs up with grass again and no amount of pushing that stupid bulb will get it to start.
I have to turn it on it's side, remove the air filter and dump fuel down it's throat, then it starts right up for about 30 seconds!
2 or 3 times and it will remain running!

Also that tiny little air filter clogs up REAL FAST and will cause running issues like loss of power at all times!

I'm still looking for an older manual choke carb to fit to that engine!