Just start rewarding supports already!!!
Riot, stick a crowbar in your damn IP wallet, pry that thing open, and starting giving players a percentage of bonus IP for queuing as support. World of Warcraft has done this for years, you should be doing it to.
You turned on auto fill so that players are sometimes forced to play a role that they don't want to. This is technically forcing that player to provide a service to the game. That service being keeping queue times low. If I'm going to provide a service, I expect something in return.
How about a higher drop rate for key fragments? Or lowering the cooldown time on chests by a percentage for every game played as support. There are ways to make players want to play it, but you won't do it because you think it will cut into your business profits too much.
You know what else will cut into your business profits? People quitting the game because of long queue times and being forced to play something that they don't want to play.