Running Proton like that is not supported. You must launch it from the Steam client.

@aeikum
do you mean that protone command line API is not supported anymore? oO

It never was. It depends on the Steam client to set a bunch of environment variables, and, in the case of 5.13, launch Proton inside the Steam Linux Runtime container.

How the users are then supposed to deal with small issues like unable to install Microsoft .NET 3 that a game requires to install? I overcome this by running proton run install.exe manually for all exe files I want to install, skipping dnet 3 because (correct me if I'm wrong) it's already included in proton wine that won't be possible otherwise for games that install dependencies sequentially and quit on the first fail? What should I do then, open an issue here for each such game?
For example, this issue: https://github.com/ValveSoftware/Proton/issues/2951

How the users are then supposed to deal with small issues like unable to install Microsoft .NET 3 that a game requires to install? I overcome this by running proton run install.exe manually for all exe files I want to install, skipping dnet 3 because (correct me if I'm wrong) it's already included in proton wine that won't be possible otherwise for games that install dependencies sequentially and quit on the first fail? What should I do then, open an issue here for each such game?
For example, this issue: #2951

I've never seen this way you show above. Always done it by using eg "$ WINEPREFIX=prefix-of-the-game winetricks ..." and as long as there's a prefix, I would expect this should work.

Edit: Indeed, using 5.13, there's still a prefix created for a game in ...Steam/steamapps/compatdata/game_store_id/pfx so you should be able to install anything to that such as dotnet3.

Then I googled the wrong usage of proton API.
I suppose, that winetricks is installed system-wide, and, in fact, I can use any wine command with a WINEPREFIX pointing to a steam compatdata drive folder like you mention, right?
In this case, you can close the issue.
Thank you.