I know it's not a great solution, I don't understand at all why after running ForgedAlliance.exe the game doesn't appear in the volume mixer and you can't change the source to some other device, when after running SupremeCommander.exe everything works without any combinations. This is totally stupid, because it's the same file just with a different name
(yes, I know this is old game but... Windows uses some audio volume mixer whitelist? Why? I don't think I'll ever get an answer to that question)
I also discovered that switching the game to the output of another device when the game is started does not work properly using the built-in Windows function.
There is a utility called AudioRouter that does this job correctly.
And forgot to mention, I'm using Windows 11.