I have an XPS 15" 9560 (with an Nvidia 1050). Some time ago, I installed Bazzite for the first time. At some point I connected it to the TV I have but there was no HDMI profile in Sounds->Speakers. I looked for solutions here and Reddit but either nothing helped or I couldn't understand it. Anyhow, I installed Kubuntu and now I was getting the HDMI Profile in Audio (Kubuntu name for Sounds) and it was working. So, I decided to test Bazzite again. I installed Bazzite and, before any updates, I got it working. In Sounds->Speakers->Profile there were HDMI options. I decided to finalize everything and update the laptop and after the updates the HDMI options were gone. So I don't know what's going on.
I have the Nvidia Legacy Drivers.
// Edit 1 //
The Problem: This is 'technically' not a Bazzite problem, though it occurs after Bazzite makes some updates and I didn't face the same issue with Kubuntu. From the little I understand, this is a pipewire/pulseaudio issue.
Partial Solution That Worked For Me: A partial solution to the problem can be found in one of the comments here https://askubuntu.com/questions/1256043/no-sound-on-hdmi-ubuntu-20-04/// It is a partial solution because it allows you to manually show another playback device that you can select to play audio from the TV (through HDMI) but it doesn't restore the user's original ability to change the profile of the built-in device/Speakers to HDMI.