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Therefore in order for the skateboard to move forward, wheel speed must be greater than treadmill speed and the conditions of the OP can't be met.

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The only way I can explain my thought process is like this. Imagine there was no ground. Do we agree that if the plane were suspended in air somehow that it could still take off via engine/prop propulsion?

Now imagine that while the plane is taking off in mid-air some force (think the treadmill) was spinning the wheels at 1 MPH. The plane is now traveling at takeoff speed and the wheels can still be spinning at 1 MPH via this force. Take this further and imagine the treadmill coming along for the ride. Treadmill going backwards at x speed, wheels spinning forward at the exact same speed. Plane flying though air at y speed.

For all intents and purposes this is what the original scenario would be like but on the ground. To observers it would look like the plane was gliding along the treadmill. Wheels and treadmill both going the same speed in opposite directions.