The reason I ask if the cap is hot indicates there is excessive starting time, or it may have to re-energize if speed decreases.
If the electrolyte solution in the cap dries up, the mfd capacity may diminish causing the cap to underperform not providing the proper boost to the aux winding.
You are correct, once ~ 80 operating speed is achieved, the centrifugal switch removes the cap from the circuit and the aux winding is de-energized.
If the cap never de-energizes it will blow or trip OL.
If it starts, I don't think a start cap is going to cause the overheating, unless the speed slows for some reason, in which case it's a secondary issue.
When a motor operates it produces work and heat. The less work it performs, the more heat it produces.
So, if the motor is oversized it will produce more heat, because the rated impeller can only produce a given amount of work at a certain rpm.
If the motor is undersized, it will not be able to maintain rpm again causing less work, and will likely cause added stress on the aux winding as well.
Both cases can cause intermittent overheating, and it can weaken the OL.
A weak OL will trip more frequently.
There also needs to be good air flow to cool the windings. If possible use some compressed air to clear any debris which can insulate the heat on the windings.
If the voltage is 240, I would check each leg to ground.
If it's 120v check neutral to ground. Hopefully you have a GFCI to eliminate this issue.