In most of the B&S valve problems I've seen the symtom is poor or no starting.
You've replaced the carb, plug, and coil. The coil and plug make up the complete ignition system on this engine. There is an "ignition kill" wire that grounds the coil output, with the ground signal coming from the ignition key and various safety switches (seat, PTO, clutch, etc.). If you remove that ground wire, and it still fails, then it's a fuel problem.
If it's a fuel problem, a possible cause is a lack of fuel tank venting. Try loosening the gas cap and run the machine for 15-20 minutes. I think I might try this first as it's the easiest to do. A partially plugged fuel line will also cause this type of problem. Try removing the fuel line from the fuel filter and blowing compressed air back into the tank (remove fuel cap).