Yes, in many formulations the unstated assumptions that the host (a) will always open a door after your initial pick, and (b) that it's always one without a car behind it. Making the assumptions explicitly makes the solution and intuition much simpler.
If your initial choice was a car, the host can open any of the remaining doors, but if your initial choice was a goat this forces the host to reveal extra information to you (namely which of the remaining doors contains the car). Since your initial probability of picking a goat was 2/3, there is 2/3 probability that the host will reveal the prize door for you.
This is why the puzzle is only loosely based on a TV show. No real TV or other iterated games will work like this, since the optimal strategy is pretty simple. In a real TV show, the host would mix up his strategy (never revealing the car door, but only occasionally opening a door after the candidates choice). In that case it's not possible to work out an optimal strategy without additional assumptions or clues wrt the host behavior. E.g. he might be biased to open a remaining door with higher probability when the initial choice was correct, to increase suspension for the viewers, in which the dominant strategy is actually to not switch. But in a real TV show or iterated game, the host behavior is likely not deterministic.