I'm the sort that does like and can appreciate the cased artifact and good copy. I tend to be a very visual learner. I also appreciate that there is such a thing as museum fatigue -- I get it too.
But this is where I believe living history museums or museums with very robust on-site education departments truly shine.
We go to a museum that is, let's say, dedicated to stone tool technology. It is not enough to simply view archaeological artifacts in display cases with copy that merely identifies the artifact as "clovis". What is it, what was it used for? What sort of economy existed that brought stone to stone poor regions? In turn if all your museum is doing is displaying artifacts with copy that only satisfies academic sorts - you suck. Such a stone tech museum should be employing educators who know how to work stone tools. It is not enough to just break stone, but to actually make stone flakes and then use them to butcher an animal.
You are a pre-industrial textile museum. Don't just put a swatch of fabric in a display case and tell me that it is a 2-in-1 twill or 3-in-1 twill cloth. Show me what it is. Explain the weft and the warp. Better yet don't just have a loom in a giant display case. There should be a working loom with someone that knows how to use it.
I would love to see more art museums with artists in residence -- as educators. Don't just create a painting, explain how and why. It is the process, not the product! Ever consider why Bob Ross is still popular?
People crave this kind of connection. This is my daily life working as a museum blacksmith. Kids especially are so mesmerized just standing there watching me make nails. I meet adult visitors telling me that they visited the museum as kids, how much fun they had, and that the blacksmith back then gave them a nail, that they still have. They had so much fun and they were now bringing their kids to see the blacksmith too. I'll give their kids a nail. Sometimes they show me the pictures that were taken those years ago and it usually takes them a moment to realize that I was the smith then too.
I had a ball visiting the Mary Rose museum where their education staff put a 1545 stone cannon ball in my hands and then a section of rope that still smelled of tar. I had an absolute blast stumbling upon the Steel Art Museum in Prague, where visitors are invited to interact with, touch, handle, even sit and climb on or into the exhibits.