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Parts for Estate EED4400WQ0 Dryer - AppliancePartsPros.com

See the attachment for the wiring diagram.

As you can see from the wiring diagram the right hand side of the heating element goes to a centrifugal switch on the motor.
There are two of them and they close when the motor gets close to operating speed.

One of them connects in the heater. This ensures that there is no heat until there is air flow.
The other one does s couple things.
It provides a parallel path around the push to start switch which then lets you release the switch and the unit keeps running.
It also disconnects the start winding when the unit is running. If left in circuit this would cause the motor to overheat.

I would check out these switches could be that the motor is clogged solid with lint so that the mechanical device in the motor is keeping the switches activated.
This would cause your heating symptom.
Also the start winding will not be in circuit so the motor would not start.

Strange that the motor does not hum but I have see stranger things.