All looks good. I can hear the motor spinning, but the drum won't turn.
Are you sure the motor is running. Often what you hear is a buzz but this is the motor trying to start but cannot.
Remove the unit from the vent system and see if it is exhausting air.
If not the motor is not starting.

I give the drum a push and it starts spinning and continues to spin.
Often this is a motor not starting symptom.
Can be a number of things.
1. The motor is toast re: the start winding is open.
2. The drum is partially jammed and the motor does not have enough oomph to start it rotating. Try turning the drum in the correct direction should rotate fairly easily. If not, could be front glides or a roller is shot.
3. The motor has 2 centrifugal switches as part of it. One of them is for the start winding. At rest the switch connects the start winding into circuit, once the motor starts the switch is activated and it removes the start winding from the circuit. Could that the mechanical device in the motor is stuck in the activated (running) position so the start winding is not connected. If the motor does not start but you have heat this is probably the problem as the second switch connects in the heater when the motor is running. Could be it is just clogged up with lint. Also the mechanism may be broken, in which case you will need a new motor.
Also the switch may be broken, again this will require that you replace the motor.

It seems like the pulley on the motor just isn't grabbing the belt to get the drum moving. Does this indicate a problem with the belt, the Idler Pulley, or is the motor starting up too fast that it can't grab the belt?
If you confirm the motor is running.
Probably not the idler wheel. Rotate the drum manually while watching the wheel if it rotates it is probably OK.
Check the belt for cracks, replace if necessary.
Remove the belt from the motor's pulley and check that the pulley and it's groves are clean.
If all looks OK then I would replace the belt.
The motor only has one speed so it cannot start up too fast.

Do belts get to a point that the get slick and the motor pulley grab correctly and can't turn the drum?
Never seen this as the ribs on the inside of the belt stop this from happening.