Why are these questions so important? If you want the objects to be "hands-on" for example for students studying these objects, they have to be properly identified. No student or researcher can work with "trilobite of unknown age and source". All collections provide hands-on research for those who request it. We do the same. I have a lot of guests coming to our collection wanting to see Triassic ichtyhosaurs, golden ammonites from Opalinus Clay or anything else. In this cases, I prepare all fossils for hands-on access.
In your case, I would recommend the Paleontological Collection of the Indiana University. The collection is quite famous, often visited, and has a very good recommendation all over the world.