First, you should learn about Wine prefixes. Arch Wiki has a good write-up about it.
After the game is installed, you need to edit that setup.exe you added as a non-Steam game and point it to the game’s actual executable.
Steam’s Wine prefixes are usually located in ~/.local/share/Steam/steamapps/compatdata. This directory should have some text config files that you can read to find out what ID Steam has assigned to it.
Also, you might have more luck with Bottles (available through flatpak) which is more suited for such tasks.






I highly suggest you stop avoiding it because it will most likely be faster and easier to do something (i.e. system-level changes) with it than not.
Similar to smartphones or MacOS, entire OS is a singular image that is also updated all at once. Core parts of the filesystem is also read-only, meaning it is pretty much impossible to mess things up if you don’t mean to do so deliberately.
The best in this regard are from uBlue project: Bazzite (most popular), Bluefin, Aurora, etc. While Bazzite is intended for gaming (things like Steam are pre-installed), the other are for general use. Bluefin uses GNOME desktop, while Aurora has KDE Plasma desktop environment. Look up their visuals and choose whichever one you like. I prefer Aurora because KDE Plasma is often much more familiar to Windows users.