As a new AoS player, I have to be honest: the biggest problem with the game is the lack of terrain rules, not the double turn. To my surprise, I've found the double turn is only really problematic with magic and shooting, and those in turn are really problematic only because terrain is utterly useless in this game. It probably worked fine back before shooting and magic were big sources of damage, but now that they are, it feels like playing 40k on planet bowling ball, but even worse, because if a shooting army gets the double turn they can literally delete an entire opposing army before it gets to do anything.
I don't see any way the double turn can survive in such a shooting-and-magic-heavy environment, unless they bring terrain rules into the game and actually allow LOS-blocking in a meaningful way.
I also don't think it's a coincidence that the most powerful armies mostly take advantage of the lack of terrain rules by going heavy into shooting and magic.
So far, my games between predominantly melee armies have been a lot of fun and feel tactical, whereas the ones with shooting armies feel like mostly just rolling dice to see who wins.