Those assumptions are required for the usual explanation. But they are very rarely stated. And so the usual explanation is not logically solid.
You can't just sneak in the assumptions. You have to state them somewhere.
well, they are part of the original, canonical problem. it's not the original problem's fault if re-tellers omit key information.
the assumption is told to you in that the game is expected to continue after monty opens a door. if he could open the prize door, the game would describe what happens when he does that