Proton is not built into Steam. It is extra software, that is a fork of wine. You could even compile it yourself, if you wanted to.
If you have Steam on your linux, you should be able to run non-Steam games with the bundled Proton.
There also seems to be ways to run Proton without having Steam installed. Using wine or other forks directly might have different results in terms of compability.
Regarding the Itch app, that is mostly a browser. It has a configuration to use a poor man's sandbox on Windows, and it caches browser games better. Otherwise there should be no difference between downloading and running games with the app or from the website, besides the launcher aspect.
So if the games you downloaded do not run with your wine, there is nothing Itch specific about it. They just do not run on that particular wine fork. Try another fork or a virtual machine. Pester the dev to release a linux build of the game. Play a different game.