As I said this is not very good behavior, some .exe files may install a program first and then it may create a desktop entry, by default this windows link will only be accessible from within the app library but you can export it. Linux uses .desktop entries, running .exe files is not how you do it.
Did you already create a bottle? If not do so, use "programs" or whatever you want to run.
pupgui is needed if you want games, install latest proton GE and use that instead with a "Games" bottle.
Then if you have any bottle created, close the app again and try to open that .exe again with bottles.
Things to consider that I have the feeling you might not know
bottles has a really well made flatpak, this is the only supported version
flatpak apps have an internal app storage (pupgui flatpak can write there to install the wine engines)
even if not in flatpak (which you should 100% use, wine is a security hole), WINE will create a fake Windows directory structure and place the .exe there
so running an .exe is completely different from an appimage (which you should not use unless you need to). This app will likely depend on Windows libraries so it needs to be installed i.e. placed in that WINE directory structure
As I mentioned in the OP, I have games installed on a separate machine and they work fine with Steam/Proton. Just click "Play" and they go. This is the functionality I'm looking for.
Several other people claimed this is how it works.
No you need some integration for sure. You could create a custom desktop entry, launching that .exe file with custom parameters. Or create or find a filemanager extension.
Simply that .exe files are often installers, bottles installs them, finds the launch links and converts them to desktop entries.
This is they way this is done. Stop using stuff the windows way, it simply doesnt work well.