Here is your parts breakdown includes a wiring diagram
Model PYE2300AZW

I am assuming that there is no hum from the motor.

First check that you have 120 volts between L1 and Neutral.
This is just a double check since the timer is running.
Put the unit into timed dry, it is just the simpler circuit.

The Current path is:
NEUTRAL , THERMAL FUSE , DOOR SWITCH , MOTOR Run and Aux windings, PUSH TO START switch , TIMER contacts M/L , L1
Once the motor starts the centrifugal switch closes 2/7 and removes the AUX (start) winding from the circuit and removes the PUSH TO START switch so TIMER contacts M/L keep it going.

Just to be sure you are getting correct continuity readings, did you remove the wires from at least one side of each device you measured? This ensures that you do not measure an alternate/parallel circuit path. Also use the most sensitive scale on your meter.

If above OK then I would check the PUSH TO START switch and TIMER contacts M/L.
If OK, check the motor could be it's internal MOTOR PROTECTOR opened and did not reset. If an open also try higher meter scale just to be sure.