They could also just tax rich people more and use that money for conservation...

They actually can't, but even if they could - expensive private viewings are much more fair.

Because you can’t just let people do their own thing right?

who is they? I doubt the Italian state can tax American or Singaporean rich tourists.

You could change tourist tax to be 5% of the room rate. That would be €5 to €10 per night for common people and €1000 per night for rich tourists.

How does this achieve the aim of a rich person seeing a painting that is not on public display?

Italy's taxes are already quite high, but they do have some regimes to encourage HNW people to move there.