If only we had some sort of state-issued document that verifies your age -- maybe even with a picture on it. I guess that's just a pipe dream, huh?
Look, I think this is a stupid law, but it's not hard to see past your objections and see where the state is coming from.
It's not all that terribly hard to get a fake ID that will pass muster at a bar. (It's a different issue to get one that will pass muster at a TSA check, or passport application, for example.)
You accidentally let a 19-year-old in to drink with a fake ID, not a huge deal in terms of liability, right? You will probably get fined if he/she gets caught in a sting, worse if they get a DUI, but it's pretty understandable and unlikely to put your strip club out of business.
But let's say a 17-year-old has a good fake ID and gets a job stripping at your club. What is your liability if someone takes pictures and you are the source of "child porn?" What about if she is doing tricks on the side and, worse than abetting prostitution, you are abetting "child prostitution?" Repeat this same exercise for any number of potential legal violations.
It is in the interest of all the strip club owners that saying "this person is OK to be a person who shows their boobies for money" is in the hands of the state rather than the bartender or bouncer who interviewed her/him on their first day of work. (And also theoretically in the interest of anyone who goes to that club and wishes to film, proposition or otherwise engage them.) It sounds puritanical at first, but from a liability limitation perspective I think it is very defensible.