In the U.S., property and privacy rights are strong. I could make a good argument for both sides. Ultimately, the issues seem practical rather than philosophical. The solution perhaps comes through government intervention, but we all dislike this. For now, taking it one step at a time seems fine.
Lets deal with the public institutions first.
The public good is really nothing more than an aggregation of private interests. So that's a difficult line to draw. But I fully agree that risk to the art works justifies limiting access. I think there needs to be a balance. A nominal charge could be levied.