Actually, if Monty chooses randomly and just happens to pick a door with a goat you should still switch, because it's still true that you will win if you switch

Actually, if Monty chooses randomly and just happens to pick a door with a goat you should still switch, because it's still true that you will win if you switch iff you chose a goat but you'll win if you stay iff you chose the car already. The host's method of selection is not relevant given a priori the observation that the host selected a goat. The host's method is relevant if we forgo that observation and iterate the game repeatedly.