But overall the system seems like the best we can get... one portion of the day with limited admission (maybe a cap at a very small number of visitors), auctioned so that those who are willing to pay the most get in. Then that money is used for upkeep & supporting general admission, where anyone can pay a low price to come in with ~no cap.
I die a little inside to discuss limiting a full experience of some of the world's best-known works of art to the wealthy, but I genuinely don't know of a better way to allocate a scarce good.