Which gets back to the question - why does/should the public support a museum. If we can't see it why are we keeping it? Even with our best preservation things will be destroyed over/with time, some things quicker than others. So if people don't get to see it what is the point of preserving it.
Museum backrooms are filled with things that they can't afford to preserve/restore, and so they are slowly being lost without anyone even able to see them in the mean time. Curators hate this reality, but they have to priorities the important things. I want things they can never preserve anyway sold the highest bidder, at least that way one person can enjoy it, we can use the proceeds to preserve something else. Plus part of the value to a rich person is showing off so there is a better chance someone will see it. (if there is no bigger that proves we don't value it. Even if future society would it won't make it to them anyway so may as well trash it now and stop pretending)
Does one get any better sense of something from seeing the original something vs a replica of the something? Does looking at the "original" copy of the constitution under all that glass do anything different than a replica under all of that glass? Would seeing the actual David statue impart any more anything than seeing a replica of it? If you say yes, why do you think any of that is the actual thing and not a replica? Just because they say so?
And don't kid yourself, those keepers and creators get full access as well as anyone they deem worthy enough. The rest of us will never be granted that access.
If it's privately funded, good. It affects me nil. But if they take public funds and lock up history or nature just so it can remain pristine for the wealthy or elite to enjoy, then I don't want to have to pay for it. Not that I have a choice in the matter either way.
That's assuming that the only point of museums is to exhibit the collection to the public. Certain museums—especially in archeology and the natural sciences—also exist to support researchers.