If the host opens the door with the car, the game would be over anyway. It is clearly implied that the host always opens a door with a goat.
"the host opens another door that he knows has a goat behind it, say No. 3,"
Using the knowledge of where the car is to avoid showing it, is more specific than just having the knowledge, and possibly still opening at random.
I think the source of the paradox lies somewhere in the biases (the Endowment effect or Status Quo bias) as discussed in the Wikipedia article about the problem - not with how the question was stated and especially not with the part about how the host selects a door.
If you model this game where the host chooses randomly, and might open the car door (when that hapoens the player instantly loses) , then there is no point switching.
I think the fact this does matter is the source of many people's confusion.
I don't think I've ever seen this rule in any variants.
Since we're making up rules: Could the player not still have the option to switch, and thus win the game?
I'm not convinced that I am wrong after your explanation.
Either way, my point is that focusing on the problem statement as the "main thing that seems to cause the paradox" allows the reader to stop there and disregard a far more interesting discussion about biases.
Why not? Your chances of choosing the correct door were 1 in 3 from the start. That doesn't change with the fact that the host opens the doors at random.
Otherwise, as a host, I would only offer the players to change if they picked the car. (And if they heard about this paradox before, they might even change their choice)
Given the description from the article:
> Suppose you're on a game show, > and you're given the choice of > three doors: Behind one door is > a car; behind the others, goats. > You pick a door, say No. 1, and > the host, who knows what's behind > the doors, opens another door, > say No. 3, which has a goat. He > then says to you, "Do you want > to pick door No. 2?" Is it to > your advantage to switch your > choice?
I'd say, no, it is not to my advantage because I'd think the host would only ask me to switch if I had taken the "right" choice and want to make me lose. Unless I knew that the host always ask if one wants to change, in which case the paradox indeed apply.
If the host does not have to open a door for the candidate (which is quite likely, he's playing his own strategy of making the show exciting, not predictable), then the usual solution (always switch) is not correct.
All this, and probably any variation and every solution of the problem that comes up in this thread (and probably every other Monty Hall forum thread in the post 3 decades) are discussed extensively in the Wikipedia entry (e.g. https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...).
Maybe every time a new Monty Hall thread comes a mod should just stick the Wikipedia entry at the top. I've never seen anything in any of the following discussion that hasn't been covered in Wikipedia already.