They did eventually open a huge replica cave which you can visit, the Caverne du Pont-d'Arc aka Chauvet 2.
Learning more about cavemen from paintings that have been studied for decades is not going to enable scientific discoveries that will advance humanity, it will not change anything to the present or the future of humanity. It literally serves no purpose other than to tell stories of a past so distant it does not hold lessons that are important for the future.
So what is left beside just the enjoyment of the public? What is the point of preserving those paintings by forbidding anyone from ever seeing them?
We have thousands of pictures of those paintings, gatekeeping their access to a handful of scientists is no different than letting them disappear entirely.
At some point the past has nothing to teach anymore.
Perhaps not the painting. Perhaps we develop a technique which can learn from the dust on the ground, or the sediments accumulated in the cave.
It is a very small cost to keep the cave closed. And in the history of humankind we have multiple times destroyed stuff later generations could have learned from.
Just look up for example the history of the Herculaneum papyri[1] and all the attempts to unroll them with destructive methods. Many of those were destroyed forever without being read or only partially read. And it very much seems like we are on the precipice of having the tech to read them non-destructively.[2]
And if all you want is to ogle at the paintings then you can do that in the replica cave?
1: https://en.wikipedia.org/wiki/Herculaneum_papyri 2: https://scrollprize.org/
The book directly pulls on the kind of data on early societies that the scientists studying these caves produce.
To preserve it a replica cave ended up being built for tourism, with the original being closed off except for scientific excursions