I currently have all three on my Mint 19 machine and want to go ahead and just remove Wine. I think it is somewhat borked anyway, and there is nothing installed there that I need to keep.
If I remove Wine via the Software Center, it won’t ripple into Proton or Lutris and cause any problems with those systems, will it?
As a test, I just renamed my ~/.wine folder and was able to run games with both Lutris and Steam Play and both ran just fine. Not sure if that is a valid test, though.
Lutris specifically depends on Wine dependencies to run Wine games. The Wine versions Lutris downloads are just the actual Wine part of everything you install when you use software manager. Think of it as “Wine uses OpenGL, so I need OpenGL installed” or something like that. If you uninstall via software manager, you’ll probably end up uninstalling these extra necessary system packages (by default).
If you really want to take Wine from your system, you can simply install every single Wine dependency by yourself and leave them on your system, but it’s much more simpler to just keep it there.
The ~/.wine folder is just a wineprefix, it doesn’t affect how Lutris, Steam or Wine stuff work because only thing installed in that container use it. For comparison, every time Lutris installs a Wine game, it creates a separate wineprefix, but instead of ~/.wine or ~/.gamename, it uses something, by default, like ~/Games/gamename as a wineprefix.
Checking Steam Proton’s page, it doesn’t mention requiring system packages to work, but if anything goes wrong, it’ll be because of missing system depedencies that can probably be resolved by having Wine installed.