As far as Steam goes, I believe there are privacy options to keep anyone from seeing your activity. Honestly, as long as you don't play multiplayer games or add any friends, you mitigate a lot of privacy risk. I do not see Linux as some kind of panacea for privacy. You have to be careful not to give away your identity or personal information, online or in real life.
I sometimes buy from GOG, but for the most part I use Itch.io. Like movies and music, I have a strong preference for independent work and experimentation.
Also, there is decades of history in gaming, with physical media or with emulation that does not require making an account or being online.