You could instead have a lottery like for Olympic Games tickets, so that only the luckiest are able to visit such places. But I don't see why the locals living

You could instead have a lottery like for Olympic Games tickets, so that only the luckiest are able to visit such places. But I don't see why the locals living there would want to. They get to set the rules, and for them the money from rich tourists is certainly more attractive than being able to "show their culture" to a balanced sample of the world population.