I think that would be a very useless thing to do. It is more culturally important to allow people to look at their history in museums. After all, we aren't talking human remains.
Problem with the museum angle is:
1. Operational costs to maintain museum and collections are constant and never-ending. Financial support is NOT never-ending in nearly all cases, and eventually you get the typical shutdown/transfer/sell/disperse activity that kills any integrity of the collection.
2. What is available for ‘viewing’ is necessarily a microcosm of the total, to limit environmental damage on artifacts, plus trying to keep an actual educational context for what is shown. Not too many people getting a cultural epiphany from masses of half-rotted flowers.
3. Once out of the environment that preserved an artifact for centuries or millenia, it begins/continues its inevitable deterioration process. Use in public displays or actively studied research collections accelerates this damage. Air pollutants, ultraviolet light, heat, cold, humidity changes - it all adds up over time.
4. The actual INFORMATION related to the items (which provide a lot of the context and value of the item) is similarly subject to loss and corruption over time - lost physical files, lost electronic/digital information.
5. For a lot of these interesting items (Egyptian, Mayan, Aztec, et. al.) that were preserved for thousands of years - they are no longer protected from exploitation and destruction. Consider the risk of a Taliban-like faction gaining a following in Egypt, and deciding all of the treasures of old need to be destroyed, gold items melted down, etc. This stuff happens, and can’t be predicted even years out, much less decades or a century or two.
Accidental fires and flooding wipes out more historical artifacts than anything else, and clearly that won’t be stopped anytime soon.
Digging stuff up to ‘learn from it and preserve it’ is just a different form of looting with the slim hope that any information gained might be preserved. I guess it lets these archeologists sleep at night.