That was the case for me when I got my PS5, I had to manually set the PS5 to being the primary, even though I didn’t even own the PS4 any more.

PS4 may still be your primary console.

Haven’t tried streaming, but I have Chromecast devices set up with the jellyfin for android TV app to stream from the NAS over LAN to any TV in the house. Basically don’t have any subscription service so these things are just Jellyfin + YouTube devices.