If it is working OK under load, that would make valve and ignition issues less likely. On the other hand, it is not that hard to get to the valves on this engine, and it never hurts to make sure they are moving freely, lap them, and adjust the clearances (0.005"-0.007" intake and 0.007"-0.009" exhaust). And clean the carbon off the head while you have it off. If you are careful, you may be able to reuse the head gasket without issue, but be prepared to replace it. If it runs OK under load, you are also getting enough fuel. The governor seems unlikely to be the problem, but you can manually hold the throttle to verify that you still have the issue.
The new carb you got may be broken. I once got one that was missing welch plugs. The lesson there is that you cannot assume that something is not the problem because a new replacement was installed. Also look for air leaks. The hardware holding the carb to the engine can sometimes work loose. If you find this to be the case, be sure to correct the problem with some Loctite, or otherwise it will work loose again.