You should be using a main source which supports the service you want to use natively. The switching solution for your audio should be completely separate and should really be considered as a solution which will likely be made 'archaic' in a matter of years, which is just the way companies like Russound and Dayton work. They make it, support it for a few years, then move on.
Another potential issue is getting a quality playback device which actually supports YouTube Music as a service. The Bluesound Node would be a product which would really work well with several solid services with great audio and solid phone app control. But, they don't appear to support YouTube Music.
If using any audio sources, that you want to play in the theater, then you need to loop that source from the DAX-88 to the AV receiver as a stereo input. If your receiver has internal downmixing of surround sound, and a zone 2, then you could loop zone 2 into the DAX-88 as a source as well.
It typically takes a few button presses to get things going if you don't have a unified control system running everything, but the native app which runs the YouTube Music app will often be the app you want to be in to run the show after music is routed to the rooms in use.
I use Sonos, and the Sonos app, with my streaming services to my home, and I use Crestron as my switching/control solution. I can pick the Sonos to play in any number of rooms without issue.
One real note: STEREO! TWO SPEAKERS!!! Put at least two speakers in each room or a single stereo speaker in the room. Your music is stereo, it comes in using stereo. Plan your entire setup around using stereo in all rooms. This will really keep things much easier overall. This is why the DAX88 has stereo inputs as well as stereo outputs on it. Keep it stereo, keep it simple.