My experience viewing La Jaconde years ago was from across the room. It was disappointing. However, at the British Museum—also years ago—the Rosetta Stone was just out there in the room next to a staircase, for anyone to molest. (I may have touched the underside of it, just for bragging rights, but I won't cop to that if asked.) And back when the Apollo 10 command module was in Michigan, I got to actually climb inside with the curator's permission. (That was a special case owing to my stature in my profession.) Now it's in San Diego behind Plexiglas.
As a former librarian, preservation is second nature to me. I've curated special collections including books from the 1500s (e.g., Palladio's Quattro libri di architettura). It's a sacred responsibility to preserve these artifacts for future generations. However, the museum experience can be disheartening. What's the point of going all the way to Paris if the closest you can get to a famous painting is a dozen meters? Why go see the Hope Diamond if it's behind a foot of bulletproof glass?
I don't know if I speak for everyone, but having these close-up experiences with history is what makes the museum special. I've been on several museum ships, for example, and I'm used to Do Not Touch signs. The preservationist in me agrees with that. But these are warships, built to last. In San Francisco harbor, the Jeremiah O'Brien has restored one of their 5-inch gun tubs to be a hands-on experience. You can sit in the seat and run the train and elevation gear and really get a feel for what it must have been like.
I really do sympathize with museums that want to put as little between the guest and the experience as possible. And I sympathize with museums that have to put Ramses II in a bulletproof glass case to keep him from being damaged by a jillion careless fingers or by some paint-throwing demonstrator. This is why we can't have nice things.