given enough time, everyone will eventually visit the switch room as many times as everyone else
I would say that this means that it will not necessarily be the case that everyone visit the room the same number of times. If the choice is truly random (say the warden computes the next prisoner from a random number table) then, given enough time, everyone will eventually visit the switch room as many times as everyone else. But if he gets everyone except #23 and has gotten #14 twice and they *have worked out a strategy (and knowing the Tappet brothers, they do have an answer that at least one of them thinks is not bogus) and the next one in is #23 then he wasn't given enough time.
From the wording it sounds like the warden can bring more than one prisoner in on a given day and can even pass several days without bringing in a prisoner so it looks like the prisoners can't even tell if the first one has gone in. This would mean that the warden could wait 30 days before bringing anyone in. The switches would then have a one in four chance of being in the orientation that they have agreed to be the sign that all 23 have been in and it's alligator lunch time.
I'd say the chances are good that, when we all tune in to hear the answer we will be greeted by the dulcet tones of Tommy saying, "Bo-o-o-o-o-o-o-gus!"
