I’m saying that by the time the wheel is rolling, the plane’s is already moving forward

The wheels are attached to the plane, so they move at the same time. There's going to be slight flex due to rubber and metal not being insanely stiff, but essentially as soon as the plane starts moving forward through the air, the wheels start rolling forward along the ground. Since the conveyor belt cancels the forward movement of the wheels, the movement of the plane ceases too.

The plane would continue accelerating even as the wheels reported weird rates of turning.

Initially, for a few tenths of a second, or a few seconds sure. But, during that time, the conveyor belt would be moving faster and faster as it matched the speed of the wheels. The faster the conveyor moved, the more friction there would be, and the more drag there would be from that friction. Eventually you'd reach an equilibrium where the drag from the wheels was equal to the thrust from the engine, and the plane would cease moving forward. It would be exactly like the plane being anchored to the ground, except instead of a stationary anchor, the anchor would be a spinning treadmill in contact with a spinning wheel. In a world without a magic conveyor belt that could instantly adjust to the speed of the wheels, there would be some slight forward and backward movement of the plane, but that's just like being attached to an anchor with a bungee rather than a rigid rope.

an affixed anchor does not allow the free motion that a wheel would.

The wheel doesn't have free motion. By definition, the conveyor is moving at the same speed as the wheel, so the wheel is locked in place. With a real conveyor belt there would of course be some lag as the motors of the conveyor accelerated the belt, but using the hypothetical as defined, the axle of the wheel couldn't ever move because every rotation of the wheel would be matched by a movement of the conveyor belt.

And one of a few things happen. Either the plane has enough engine thrust to overcome the acceleration induced by the wheels, and therefore takes off, or it does not.

The thrust would have to be infinite because, by definition, the conveyor is always going to match the velocity of the wheels. If the wheels were truly frictionless, then the conveyor belt would have no effect at all. But, any real wheel will have some friction that will increase with speed, so there will always be some speed where the force backwards from the friction of the spinning wheels matches the force of the engine.

As an aside, my guess is that most real airplane wheels would probably fail pretty quickly at just double the normal takeoff / landing speed. The centripetal force acting on the spinning parts of the wheel and tire increase with the square of the velocity, so 2x as fast means 4x as much force. 3x as fast and 9x as much force. So, if you did this with a real wheel, you'd destroy the wheel pretty quickly. Of course, the same applies to the conveyor belt, but I'm going to assume that it's specially engineered to survive this challenge.

the wheels would continue spinning in increasing RPM until the plane begins moving backwards

The plane wouldn't move backwards because if the wheels slowed down, the conveyor belt would slow down too. Of course, that's in a world where the conveyor belt could adjust its velocity instantaneously, but for this thought-experiment you can say that if the pilot cuts the engine or something, the wheels don't spin as fast, so the conveyor belt slows down, and the plane remains in one spot.

eventually it would take off anyhow. because the airflow over the wings would still generate lift

In the thought-experiment world, there wouldn't be any airflow over the wings because the plane would be stationary. In reality, there would be some airflow due to the movement of the conveyor belt, but the wheels would probably melt long before that was enough air to give the plane lift while stationary relative to the world around.