With my configuration, it does. Though it likely also depends on GPU drivers, Monitor, etc. I’m just speculating as I have no clue beyond “works for me”.

Any games that have an HDR option in their graphics settings would support HDR I hope. Maybe if the display is already in HDR while in desktop mode, it’s possible that launching the game isn’t able to toggle it off?

Washed out and desaturated is the opposite of what it should do. Sounds like you may be looking at SDR (non HDR) content with HDR enabled?