I use
Group Synergizer to help with knowing when a queue has "popped".
Unfortunately, it doesn't toggle your game's sound on automatically when the queue pops. You'd have to have the game's sound on.
But... it seems likely that it's something that could be implemented into the addon. I imagine there's some kind of event an addon could look for, and once it happened, it made sure the game's sounds were turned on.
Maybe leave a request for that on the Group Synergizer's comment page? I really like your idea, too.
(Unfortunately, the author hasn't been logged on to ESOUI in over a month. But hey, you never know!)