Play Normal Draft or Ranked instead of Blind Pick. You already CAN do this in Draft Pick.
And before we had role selection in Draft Pick, we had Team Builder (first created in 2013? 2014? I don't remember, one of those), which allowed you to select a champion and position to queue as or create a group. People who queued would find the groups belonging to the people who created the groups, and the group leaders would accept or decline them. If you queued and got accepted to a group, it would pop up a ready check on your screen. You could also leave the group and requeue at any time, so if you decided your teammates were dicks and you hated them, you could leave without getting a dodge penalty. When a group had all 5 members, the party leader started a search for another completed party, which would usually take like 10 seconds. You would then enter the game, having had the ability to customize your runes and masteries at any time before you found the enemy team.
Team Builder had a couple of problems, though. Primarily, mid laners would take 20 minutes to find a group if they didn't make one, meanwhile supports were in such high demand that they would take under 1 second to find a group and people would take any support they found.