LFG is horrible, but a solo queue would require far more than you’re aware of.
There’s the problem with emotional outrage, which is less inhibited thus the design needs to put in huge punishments just to separate such people and scare off people from acting like that. Which every single game with a rated solo queue has had to find ways to deal with, usually by having a strict punishment system in place.
Then there’s the problem of how you’re locked into one character, which they’d need to provide arena tournament realms, queuing from interface and the ability to set up the UI from somewhere for each class/spec, and a lobby where you pick what to play after having been matched up with ready-made characters without need for leveling or gearing.
Since that’s an auto-regulating system so you’re not locked into only playing that one thing, so people will fill the role they need to fill to make it work with the people they’re matched up with (also makes it possible for “role” queues).
But the main problem with the group finder is that people are using it to find groups way too much. It creates the illusion that the grass is always greener on the other side, with the other side being to find a new player whenever a problem comes up. It has degraded over years of decay, into the state it is now.
Which Suggestion to improve the group finder for rated PvP helps with.