Wouldn't the jets being loose create the over fueling problem? With many Nikki carbs the main jets are held in place by the mini o-ring when they get hard the flow of fuel moving thru the nozzle will actually lift the jet out of place and more fuel flows around the bad o-ring causing the over...
Folks we can blame the CEO, the stock holders or who ever we want but the American buying public is to blame for 2+ decades the cry has been "We want it cheaper" and manufacturers have done just that. Say thank you to China when you buy your next mower.
Sorry if this has already been said, but you have compression issue trying readjusting the valves if that doesn't work you probably have a bad camshaft that's not working the decompression release. Good luck.
What's the engine model? Check and see if you are getting 12v to the control wire at the base of the solenoid, if you are check for 12v on the starter side of solenoid.
When it shuts off the first thing I would do is check for spark on the ignition coils and second see if the fuel solenoid piston is pulling down and staying down as long as power is applied.
My thoughts go a weak ignition coil or fuel solenoid, i'd suggest when it shuts off check the plug wire for spark and see if the fuel solenoid is pulling the plunger down.
What are the moisture conditions when you mow? If the grass or ground has much moisture in it this also will contribute to plugging, and I agree with other the high lift is the way to go for bagging. Good luck.