You probably need a new motor, sounds like the start winding is failing. Your description of hand starting it is a classic diagnosis technique. If you can continue to get it to work this way there's no issue. The motor has a thermal safety that would power it down until it cooled if you let it hum for several minutes.
As long as the drum turns easily by hand I'd say motor. Objects stuck in the blower can mimic the issue but usually make noise.

Other than getting the blower wheel off it isn't a hard repair. It isn't uncommon to have to break the blower off the old motor, and the plastic blower is very tough. A 1/2" drive ratchet should remove the blower, and it is probably reverse thread but it's 50/50 on whether you can get it off the proper way. If you have time to repeatedly spray it over a few days with a penetrating oil such as WD-40, LiquidWrench or Aerokroil you may be lucky.

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