Check out this old post: https://www.realtimerendering.com/blog/thought-for-the-day/
HDR in games would frequently mean clipping highlights and adding bloom. Prior the "HDR" exposure looked rather flat.
OK, so it doesn't mean real HDR but simulated HDR.
Maybe when proper HDR support becomes mainstream in 3D engines, that problem will go away.
That's not what it means since 2016 or so when consumer TVs got support for properly displaying brighter whites and colors.
It definitely adds detail now, and for the last 8-9 years.
Though consumer TVs obviously still fall short of being as bright at peak as the real world. (We'll probably never want our TV to burn out our vision like the sun, though, but probably hitting highs at least in the 1-2000nit range vs the 500-700 that a lot peak at right now would be nice for most uses.