To address your edit, you can actually add non-Steam games to Steam. I’ll provide a link that describes the process, but you may have mixed results (though you may be surprised how many of them “just work”):
You’ll find instructions in the section labelled “Can Steam Play be used for games not on Steam?”. I believe these instructions still work, even though they’re a few years old. If not, then there’s always the option to use Lutris or Glorious Eggroll, which I recommend looking into yourself if this solution doesn’t work (Lutris should be pretty easy to set up, so I recommend trying that first as I’ve had a good experience with it).