"And this would be some heuristic which orders people in an unambiguous way, so only one person is attempting to exit at a given moment."
Given how badly people behave at four-way stop signs (where there's an unambiguous ordering algorithm that most drivers seem to ignore), I don't think there's much hope for a heuristic ordering working well for panicked crowds at emergency exits.
Are you sure it's unambiguous? Four way stops have both a "whoever was here first goes" rule and a "we arrived at approximately the same time so the person on the right goes" rule. That alone adds ambiguity, as they're not well defined.
But it gets worse than that, as pedestrians have priority and can break up the established order. If it's "my" turn, but a pedestrian blocks only my way, someone else can rightly cross out of turn, but then once the pedestrian is clear when does it become my turn again?
I moved to the US four years ago from the UK, and I think four way stops are wonderful. I've yet to experience the disorder suggested. I grip the wheel, narrow my eyes and approach with extreme caution. I like think everyone else does too.
Given how badly people behave at four-way stop signs (where there's an unambiguous ordering algorithm that most drivers seem to ignore), I don't think there's much hope for a heuristic ordering working well for panicked crowds at emergency exits.