Outdoor Power Equipment and Small Engines - John Deere backfires

I have a early 1970's John Deere 214 garden tractor with a 14 hp kohler K321AQS engine. I can start the tractor fine but when it runs at full throttle for about 10 min. the engine will backfire and die. I can start it back up right away, run it for 5 min. or less and the same thing will happen. The only way I can keep it runnning is run the tractor maybe half throttle or less. A friend suggested changing coil condenser and points. I have done the first two. Any ideas? One person said it might be a sticky valve. Any help would be appreciated.

I would continue and replace the points as well (easy enough to do and needs to be done from time to time anyway even if they aren't the problem). Also, I suggest loosening the gas cap and see if it still does it. If the gas tank cap vent is not working, the gas tank could be developing a negative pressure, keeping fuel from flowing to the carb.

A sticky valve is possible, but I doubt it. The fact that it starts right back up immediately could make it hard to diagnose. Does it do anything odd just before it backfires? (surging, sputtering, etc...)