You can also use Steam as a launcher. In Desktop Mode there is a menu entry “Add a Non-Steam game to my Steam Library”. For Windows games, you can just browse to their .exe file. After adding it to the library, you can open the Library Entry’s Properties page, and choose Proton as compatibility tool.

That way you get your non-Steam games in your Gaming Mode launcher.

To get nicer images, there’s a website named www.steamgriddb.com that also has a small Flatpak tool that you can use in Desktop Mode to set icons/banners for your Non-Steam games.