Here are your parts includes a wiring diagram.
Parts for Maytag PYET444AYW Dryer - AppliancePartsPros.com

I cannot figure out how this unit is doing this as the door switch removes Neutral from the motor when it is open.

Is this unit new to you and/or hasit ever run correctly?
My thinking here is someone has messed with the wiring.

How did you check the door switch?

For it to run the motor must be getting Neutral onto connection 1 which the door switch should disconnect.
Perhaps there is a short in the motor to ground onto this connection.

Hopefully you have a meter.
Set the meter to the most sensitive resistance reading usually 200 ohms.
Before starting short the meter leads together so you can tell if there is a zero offset in the meter.
Unplug the motor now check 1, 2 and 7 to the frame/ground.
All of them should be infinite ohms.
If not there is a short in the motor probably to ground so the motor is getting power from L1 to ground instead of L1 to Neutral with ground just being an electrical safety.