Toss a coin and I'll tell you what happens allows someone's action to matter and to have meaningful consequences.
But it doesn't give that person agency over the game. All the decisions about what happens next are being made elsewhere.
Which is @hawkeyefan's point.
It is possible to have RPGing which has more agency than a blind coin toss, and that doesn't involve the players just making things up like "whose line is it anyway". But to talk about that sort of RPGing we need to talk about techniques, and systems, that go beyond state-of-the-art circa 1984.
I don't think it even does, honestly. As I noted, if I know the costs and benefits possible, have some way to be able to assess them, and can do things to put my thumb on the scale, I have a serious degree of agency, even if I can't walk away from the decision entirely. The less true that is, the less agency I have there.
That's why the old two-doors example is so agency free; until more options are provided to assess the doors, providing me those just makes me a human randomizer; there's nothing meaningful there in the way of agency. I don't know if the doors matter at all, and if I have reason to think they do, I might as well flip a coin myself. This can be changed by any number of elements in the situation, but those elements have to exist, and among other things, the greater degree of blind decision the GM wants, the less, in the end, they really want any meaningful agency.