But Mean Girls and Frozen do not need a Tony to keep running, neither did Wicked or Phantom, or Lion King, etc. The Band's Visit has SURELY run much longer and gotten more interest from ticket buyers because it was the Tony Winner for Best Musical... and I'm sure the same is true for Avenue Q... and I'm sure it would have proven true for The Great Comet, and Ragtime and Into the Woods and Sunday in the Park and countless others that absolutely should have won and didn't.
I've never understood the logic behind giving the one award that almost definitely can greatly improve box office to a show that not only does not need it to help but isn't actually the best musical in terms of quality that season. I'm very glad Avenue Q won over Wicked because I think it's a better and more original show - both are good, and I would have given it to Caroline or Change that year, but I think giving it to the underdog was the wise idea because Avenue Q had an amazing 6+ year run on Broadway for a show I doubt anyone expected to run more than 2 years. And Wicked, without the Tony, has been running for 14+ years. There's no benefit to giving the presumed winner the win if the presumed winner's box office doesn't need the help and it isn't a better show.