Because the signal is carried via the HDMI ARC port on the TV and if the TV is off, so is the port.

system with volume control runs fine with music via the app and the tv is off.

If you have an infrared enabled ‘remote’ about the place, just add that via the Arc room settings to control it’s volume whilst the TV is powered off - that’s what I do with my cable TV remote.

A bit irrelevant, as the Sonos App/controller device is connected over WiFi and is designed to control the Arc’s volume.... you are best to add an infrared remote to the Arc room settings… see this link (ignore the part about the optical connection):

https://support.sonos.com/en-us/article/check-if-you-have-an-ir-remote

Remote works arc volume fine with tv on.

Remote does not work arc with tv off and music playing (sonos radio)

Read the replies, it contains the reason why.