Air conditioning fan not turning, motor very hot
Your idea that the capacitor is bad is probably correct.  With a bad capacitor, the fan can't kick off and start turning, but there is still electricity running through it.  No fan turning means no cooling so that heat builds up.  Having a capacitor go bad unnoticed can actually damage the fan as well because of this heat.

You tried turning the fan, which is a good troubleshooting step.  Since it does turn, replace the (inexpensive) capacitor.  If the motor doesn't come on after that, it could be damaged.  You could try to check the resistance of the motor coils, but you'd need to find a reference for your fan motor to know what "good" was.