I am by far a small engine expert, but I have seen this before
Is the flywheel tight enough?
Is the taper worn on the shaft or the flywheel?
Is the flywheel hitting something?
Does it have any kind of a load attached to it? Like a belt accessory?
I assume you are using the Kohler part?
Pull start or electric? Is the flywheel hitting the starter or is the starter drive not completely disengaging?
Food for thought until someone better with these comes on.