As far as solving the problem in an academic sense, your explanation is clearer. However, one of the things that makes the whole thing fascinating is that the actual players are not told this specific information, that the host knows. If you watch the show religiously it might dawn on you, hey, he must not be picking the preview door randomly because he has never accidentally revealed the car -- but that's beyond most people's observation.
In practice, the revelation of one door is likely to make the player even more committed to the original choice -- the odds (seemingly, not actually) just went from 33% to 50%, momentum is good, I'm feeling lucky, no way am I going to switch now!
The "Big Deal" -- the game where the player chooses one of three doors -- was the finale of every episode of Let's Make a Deal under Monty Hall. Still is, in the Wayne Brady revival. Not hypothetical.
> The show's host then reveals the contents of one of the two remaining boxes, but, importantly, always a "goat box."
but it could be made slightly more explicit as:
The show's host, who knows the location of the car, is then compelled to always reveal one of the two unselected boxes as containing a goat.