You can distill it all to this:

If your initial guess is correct, then switching always loses.

If your initial guess is wrong, then switching always wins.

Everyone who understands the game can agree on those facts without controversy. If they don’t then there’s no point proceeding because they do not understand the rules of the game.

Therefore, the odds of winning when you switch are the inverse of the odds of winning from your initial guess.

If your initial guess wins 1/3 of the time, that means always switching loses 1/3 of the time and conversely always switching wins 2/3 of the time.