I understand the false start about taking the wheels off, ... had plenty of false starts myself, so no explanation is needed.
I would question, "... wheels are being driven ...," and "... stop the right wheel ...." If the right one is being driven, you should NOT be able to stop it with your foot. That would tell me that the wheel is not being driven with power. Most likely, the outboard drive shaft is turning, but the key-driven pinion gear is not locked into place with forward motion. The outboard shaft has an inset, that is a square recess into the shaft near the end. It is about 1/8" wide, and 3/8" long. A small spring is set into this recess, and a key is placed over the spring.
The key engages the inside of the pinion gear. The inside surface of the pinion gear has three engagement points for forward operation only. The engagement slot has a hard angle on one side the of the slot, the other side is sloped. This allows for the gear to rotate backward when the operator pulls the mower in reverse.
My guess is that something is wrong with the pinion gear engagement with the outboard driving shaft. The key is not engaging the pinion gear, so that the pinion gear is nearly full floating. When you raise the rear of the mower, there is enough friction to make the pinion gear turn the free-wheeling wheel. But, when resistance is applied (e.g. your foot to the wheel), then the friction will no longer drive the wheel. The outboard shaft is rotating freely inside the pinion gear.
This is my first-best guess as to what is happening. If none of this makes sense, let me know. I have some pics of the pinion gear assembly. (I just changed out the lever arm to holds the rear wheel -- have yet to get some annotations, but intend to provide a similar tutorial on how to make this repair).