In fact, by my wording, there's a logical incosistency if you use a car. Car moves -> Belt moves other way -> Car's stationary -> ERROR. Which is fine, because it's a plane - not a car. The question phrased as I have it still tends to lead people into the dyno/car comparison and they make the same mistake.
My biggest trouble with your wording is that, although I feel what "at the speed the vehicle would be doing if the belt weren't there" means, I'm not sure it can be precisely defined. I haven't followed it through, but worry it might be open to contradictory legitimate interpretations.
The problem as worded in the original post of this thread is appalling though.
...designed to MATCH the speed of the wheels of an airplane...
Firstly, define "speed of the wheel". Is that "the rate at which the axle of the wheel gets from one place to another", or "the reading on a speedometer connected to the wheel". They are quite different. A wheel not in contact with the ground that has been fired from a cannon will have a high speed BUT as it isn't rotating a speedometer connected to it will display zero.
Conversely a motionless wheel on a moving treadmill isn't going anywhere (so has no speed), but a speedometer connected to it will show a non-zero value.
If "speed of the wheels" in the question DOES mean "how quickly the wheels get from one place to another" then, because they're connected to the plane, it means EXACTLY the same thing my phrasing of the question does.
If instead "speed of the wheels" means "reading shown on a speedometer connected to the wheel" then the question BY DEFINITION (literally) says that the plane cannot move.
ie. A speedometer on a wheel touching a conveyor belt will show the "speed at which the wheel is getting somewhere else" PLUS "the speed at which the belt is moving under the wheel". So if the question defines the belt as one that moves at the speed shown on the wheel's speedometer, the "speed at which the wheel is getting somewhere else" is DEFINED as zero. As the aeroplane's attached to the wheel then its speed is ALSO DEFINED as zero. There's no mechanical process holding the aircraft there, just the phrasing of the question has effectively become: "There is a belt with a motionless aircraft that will always remain motionless on it. Can the aircraft move?" Obviously a trivial question, and so I reject the possibility that the question could mean that.
EDIT: Written as the last four posts were made. Everything since Rey's.
EDIT2: Added philip to start of post.
EDIT3: Added "that will always remain motionless" to last paragraph.
EDIT4: Added EDIT2, EDIT3, & EDIT4 notes.
EDIT5: Changed "the rate at which the axle of the wheel moves" to "the rate at which the axle of the wheel gets from one place to another" and EDIT5 note.