I agree with everything above, but still why off/on appears to cure it?

The engine shakes on stop and start, releasing the jammed part. Nothing here indicates something other than a mechanical problem if the car does not move when apparently in gear, engine running, no foot on pedals.

Brakes and clutch?

I think I would try removing the pedals(first) and inspecting them/cleaning/freeing off/lubricating, then inspect as others have said the cross linkage for the brakes(drivers side to passenger side under the dash.

My concern is the apparent lack of brakes when the clutch issue happens and that it may, possibly (at some point) not brake when required(rock hard pedal) whilst normally driving?