Grab a Steam installer from store.steampowered.com, and run it in Wine. Go to where you downloaded it, open a Terminal, and type wine and a space into the terminal. Drag and drop the Steam installer and start installing Steam.
Also, switch back to the apt installed wine-staging when you have to install a new application in Wine. The Lutris builds are based off of different path variables compared to your apt installation so icons and paths could break if you install new programs with the Lutris pre-build. To be on the safe side, switch back to your PPA sourced wine-staging installation when you have to run a installer.
If everything goes right, it will not crash past the splash screen and you will be able to see the Vicarious Visions intro. At this point, if you can see the Vicarious Visions intro, plug in your controller. Leave your controller disconnected when launching the game each time, THEN connect when you can see the Vicarious Visions intro. This ensures the controller works correctly. You can then skip to the logo, and exit the game.
Congrats! You can now play NST using Vulkan on Linux and you can report your performance playing the game to whereever. Currently Cold Hard Crash is the laggiest map even on a GTX 1080 running Vulkan if you use the release DXVK version 0.64, but if you use DXVK 0.70+, performance is completely gained back and is on par with the Windows version.