Remove the valve cover and disengauge the exhaust valve rocker or take out the exhaust valve push rod so as the exhaust valve stays closed and do the compression test. This will dis-able the compression release. On a OHV engine you can do this by just pushing down with finger pressure since the valve springs are weaker as compared to a L head valve spring. Do not let the push rod fall into the block.
While you are in this area check the valve clearances.
Also adding some choke
at idle and the surging and hunting decreases indicates the engine is running lean. (too much air or not enough gas)Some of the idle jets have several small cross drilled holes and a center hole. All the little holes have to be very very clean. Just one of the itty bitty holes restricted will cause such. May have to use a torch cleaning tip tool or carb type jet cleaning tools instead of just air cleaning.
Send KE4ABB a PM about his recipe for resizing just a itty bit. If you don't find valve issues.
I've done such also to some carbs with real good results, but nuff said.
I've also seen some of the low emission Briggs carbs with a low speed adjustable idle jet that would idle OK but hunt and surge at
medium throttle or off idle speed and just a 1/8 turn of the idle jet would correct the H roughness or surging. (then check the critical L adjustment at both cold startup and warm engine)