This is a feature missing from current Big Picture. Being able to choose your computer's audio output would be extremely useful. My own setup could benefit from this greatly, and currently I am limited in audio devices because i do not want to get up and manage my devices every time I want to switch. Having this built into Steam would be amazing.
Maybe Im wrong, but this used to be a feature in old Big Picture... does anyone know if that is true? This is just based off of memory.

Also.. this already exists on Steam Deck. The exact implementation that would solve my audio setup issues is already here. Why is this not a part of desktop big picture on PC?

I'm aware of various third party audio management solutions. They are not very good solutions.