The old joke about what they call a chicken fucker in Oklahoma comes to mind here except I think those pre-Turks were cavorting with leopards.
Maybe the ones who built this site went to the Spanish/French border and are the ancestors of the Basques, who have a unique language. I'm not the guy to answer that question.
If I wanted to identify this group's descendants among modern people I would look for people who have a deep connection to cats like the ones depicted in their monuments.
And now you know why I am not much of an archaeologist. I get a little tired of every polished rock being something someone worshiped though it looks like for these people that probably fits.
Technically, everyone could be descended from their closely related neighbors who got sick of the cat people, rather than from the cat people themselves, though.