Actually that is incorrect as there is no guarantee that every prisoner has been in the room at least once over any amount of time. Of course the probability will get higher and higher that everyone was in the room once if it was completely random but that probability will never be 100%.
I know of two legitimate answers to this problem, it took me awhile when I first heard it.
IIRC, in the original statement of the problem, there's also a stipulation that, for all prisoners, the king/chooser/whatever will visit them an infinite number of times (so at any time it must be true that each prisoner will be visited [again or for the first time] if the game doesn't terminate).
So, if you wait a finite amount of time (in this case a "few" billion years") it's possible that the visiting condition might not yet be satisfied.
This is correct.
> The solution only requires that they will eventually be visited; you can keep deferring until they are.
This is false; you can't "keep deferring until everyone is visited" because you have no way of assessing whether that's happened. As a strategy, it is impossible to implement.