When you use the kick starter, does the motor feel about the same to kick whether warm or cold? I would expect it to be slightly harder when cold, but hard to tell much difference.
This test should indicate whether the problem is with the starter motor system and its drive or in the engine itself.
Is the engine definitely spinning fast when cranking cold or could the starter be spinning without driving the engine? If this is the case it would have to be the starter sprague clutch.