I don't want to argue which rule would be more natural or make a more interesting show. But without a clear rule, I cannot decide on a playing strategy.
> Why "provided he gets into the gameshow final"?
Because I assumed that the host always has to open a door and give a choice to switch.
It doesn't make any difference to your strategy! You don't get any choice if the host opens the door and reveals the prize, so your strategy can't possibly be affected by what the payoff for that scenario is - even if you win 10x the prize if that happens, or get executed if that happens, it makes no difference to whether you should switch or not in the scenario where you actually do get given a choice.
But if the rule were to reset the game until the host doesn't make any more mistakes, it matters.
The most natural rule is that the guest loses: the guest picked a door that didn't have the prize, they get what's behind "their" door. (And note that the much-argued formulation already "changes" the game from what was done on the original gameshow).
> The problem statement is: "What strategy (stick or switch) is best for the guest (= has higher probability to win), provided he gets into the game show final."
Why "provided he gets into the gameshow final" and not "provided he gets into the gameshow final and is offered the chance to switch"? Like, if you're asking whether a chess player should take an en passant capture, you'd look at whether capturing or declining en passant leads to winning more often, you wouldn't look at all possible chess games (including those where there was no chance to capture en passant) because that just adds a bunch of irrelevant cases.