[rmacqueen]

I swear this problem only ever causes confusion because the original wording is ambiguous about what the host's motivations are. Once you realize the host is trying to get you to lose and knows what's behind each door and must open a door, then it's clearer that you should switch.

[kergonath]

The host’s motives don’t matter at all. If the first choice is a goat, then the host does not have any choice about the door to open, so their intent is irrelevant. If the first choice is a goat, then the outcome is the same regardless of the door the host chooses.

The host does not really have any agency here.

The rules are intended to confuse people and trick them into thinking both remaining doors have the same probability. I can see how this would favour the house on average because once people committed and chose a door, they are invested and would tend not to change their mind. But the host’s actions have no effect. And even then, this might be an intuition, but does not provide any understanding as to what exactly happens.

[rmacqueen]

I'm not sure if you would call it 'agency' but the host is following it's own specific door-picking policy - a policy which will never result in a car being revealed. It is precisely this policy and no other which makes the contestant switch the correct move. But the original wording doesn't say that - it just says the host opens a door and it happens to have a goat behind it.