Just go to the store page of a game that has a demo, and tap/click "Download" where it says download demo.
You mention "the library is shared between Linux and windows", which is concerning.
If you are mounting an NTFS windows partition inside Linux so you don't have to redownload your games, that won't work.
Running games installed onto an NTFS partition is really fiddly on Linux, and steam WILL break things when you try to run games installed on linux in windows, and vice versa.
You need to set a default compatibility tool for games that don't have one. I don't remember exactly where to find it and don't have access to look right now, but it's not really hidden in the settings menu.
I had the same issue once. I suspect that whatever version had been the default was removed for one reason or another, even though I never manually remove them.
You need to enable proton for all games. It's in Settings>Compatibility.
NTFS doesn't support the permission structure used by linux, so while it can work, when set up correctly, your system has to do a bunch of extra processing on top.
Not to mention that both NTFS modules available, take a performance penalty in general.