Sounds like the motor is "humming but not starting" when power is applied to the motor by pressing the start control.

These are split-phase motors; which means that the motor has mechanical parts inside it to engage and disengage the starting circuit. Normally with a dryer, the motor is powered on once and allowed to run for 30 - 45 minutes uninterrupted. Therefore the starting mechanism "wearing out" is not very likely. 

The more likely problem is that the motor's bearings have become dry of oil, and suffered severe wear. Due to this severe wear, the shaft now has up-and-down movement. With the worn bearings, the parts inside the motor can touch each other. Because of electromagnetism, this locks the motor when power is turned on. More than likely, the motor will turn by hand while the power is off; but will lock when the motor is powered on.

With the belt removed, see if you can move the shaft up-and-down, or side-to-side with the power off. If you feel play in the shaft, this is your problem. If there is no play in the shaft, and it turns free and smoothly - you may need to investigate the centrifugal mechanism or the motor's start-switch. 

The start-switch is the brown terminal block part at the rear of the motor, where the wires connect. Underneath it is a mechanism which presses against the switch contact when the motor is stopped and retracts into the motor when the motor is running. Inspect this for any sort of dust, gumming, or other problems that would keep it from cycling properly. 

Sincerely,

David