But when native games do that it doesn't bother you? With Wine at least I can disable this behavior with few clicks in winecfg to disable access outside wineprefix.

Yes, it does. In fact, it bothers me even more. I have ways of dealing with the rare cases when games write things to places I consider inappropriate, even with wine. In fact, I have to do special things for the HBS games I recommended above, since they try to write to the game's installation directory, and I don't allow that. It doesn't change the fact that Wine encourages this by default (and, in fact, used to not do this, a long time ago). Probably the only reason I brought this particular issue up is that it's one of the reasons I no longer play games much after switching "back" to Debian: I can't patch stupid stuff like that out of wine any more, because I have no idea how to build 32-bit software without creating a 32-bit chroot for that purpose. I don't have the patience for that any more. In Gentoo, I just added my patches to the appropriate place and kept them up-to-date. But I don't have the patience for Gentoo's BS any more, either.

A couple games that are sold here as Windows builds only, but can actually be played natively on Linux:
- Alpha Centauri, using Loki’s native engine - https://downloads.dotslashplay.it/games/alpha-centauri/alpha-centauri-linux.tar.xz
- Caesar 3, using Julius - https://github.com/bvschaik/julius/releases
- Diablo, using devilutionX - https://github.com/diasurgical/devilutionX/releases
- Heroes of Might and Magic 2, using fheroes2 - https://ihhub.github.io/fheroes2/
- Heroes of Might and Magic 3, using VCMI - https://vcmi.eu/
- Jazz Jackrabbit 2, using Jazz² Resurrection - http://deat.tk/jazz2/
- The Elder Scrolls 3: Morrowind, using OpenMW - https://openmw.org/
- Theme Hospital, using CorsixTH - https://corsixth.com/
- Unreal Tournament (no longer sold here), using OldUnreal’s native engine - https://github.com/OldUnreal/UnrealTournamentPatches/releases
- Unreal Tournament 2004 (no longer sold here), using BeyondUnreal’s native engine - https://liandri.beyondunreal.com/Unreal_Tournament_2004#Essential_Files

It’s been so long that I don’t even remember the in-game music (beside of course the menu music)…

This game truly deserves a modern engine, maybe some kind of remaster, but sadly I have seen nothing like this yet.

Shiiit, I didn't know that one. R.I.P. backlog again I guess.

Few more to the list:
Jagged Alliance 2 - Stracciatella
Arx Fatalis - Arx Libertatis
Nox - OpenNox
Gothic 2: NotR - OpenGothic
Freespace 2 - FS2 Open
Dink Smallwood - FreeDink
Call to Power 2 - Apolyton

A few favourites of mine:

Broken Sword
Aragami
Tetrobot & Co.
Shadow Tactics
Day of the Tentacle
Beyond a Steel Sky

I remember this open source re-implementation of the RollerCoaster Tycoon 2 engine:
OpenRCT2 - https://openrct2.org/

Also, since they share the same engine, I remember there was a version for Locomotion. OpenLoco, I think it was called.