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Parts for Kenmore Dryer 11067522600 - AppliancePartsPros.com

See the attachment for the tech sheet.

You are right that it is weird but should be fixable.

Can you localize where the hum is coming from?

It could be that the belt is broken and the hum is the motor running.
The reason is that if the heater circuit is OK then it should not heat up unless the motor is on. You will see a centrifugal switch(2M/1M) in the heater circuit path in the wiring diagram. This switch closes when the motor gets close to operating speed and ensures that the heater does not come on until there is air flow.
Of course assuming that the heater circuit is OK may not be correct.
Also this does not explain why it start up on it's own.
The reason I say the above is that you said it was a faint hum.

Below is an alternate theory but usually the hum is fairly loud.
This is the most likely cause.

You will also see a second centrifugal switch (5M/6) on the motor. This switch also is supposed to activate when the motor gets close to operating speed. It does two things. It disconnects the motor's start winding. If left connected while running it would overheat the motor. It also provides an alternate/parallel circuit path for the push to start switch. This lets you release the switch once the motor starts and then keeps it running.

Now lets say that for some reason (could just be clogged solid with lint) that the centrifugal switches are stuck in the activated position.
There will be heat, it will try to start without the push to start being pressed and the motor will not start because the start winding is disconnected.

If the centrifugal switches are being activated and cleaning the motor does not solve the problem then you will have to replace the motor.

Since you are tearing the unit apart I would check the belt, seals and other parts for wear etc.
Also a good time to check/clean your vent system.