A question about the aeroplane on a treadmill. Can you solve this conundrum?

Now we all know that theoretically an aeroplane on a treadmill will take off, as the engines apply the force to the air, not to the wheels.
The problem I have is, since the treadmill is supposed to match the speed of the wheels, but because the plane will move in the treadmill’s reference frame, the wheels will invariably be faster than the treadmill at any given time. This means, assuming an infinitely powerful treadmill, would nearly instantly accelerate it’s belt and the aeroplane’s wheels to the speed of light, causing massive friction and by that heat instantly, vaporising the treadmill and aeroplane immediately. Which means the aeroplane will not take off.
This is what I came up with. What is your take on this scenario?