I would think it would show that on an inertial dyno but not a load dyno. Not meaningless but different
Exactly this. If your dyno is measuring power output directly (water brake, eddy current, electro dynamic, or even a Prony brake), then there will be no difference in measured output between wheels of different masses, because the dyno is making a steady state measurement of torque, so the rotating masses have already been accelerated to the speed of measurement.
If your dyno is an inertia dyno, you're not actually measuring torque (or power output) directly. You're measuring the time the driveline takes to accelerate a large rotating mass to some maximum speed (storing kinetic energy in the roller). Anything that changes the total angular momentum of the system will change the rate of acceleration, and thus will change the measured power. I'd call it an error in reporting if the difference wasn't entered into the calculations.
In reality, the engine's power output is the same
with both wheels and both dynos. The difference is how much kinetic energy you're storing in the system in the inertia dyno.
That said, yes it's an immediate performance gain for a car you want to accelerate and decelerate quickly to have lighter wheels because there's less angular momentum to store in the wheel when accelerating, and less to dissipate when decelerating. The car will handle and ride better too, because unsprung weight is the enemy of good handling and ride quality.
It's yet another reason that Chapman was fanatical about weight.