So the main reason why there's no normal draft mode for TT is that there currently aren't enough players to split them between two different normal queues without severely impacting the time it takes to match with other people or the quality of the matches that get made. We're not against the idea, it's just not something that we think would improve the quality of the games in 3v3s, since you may get to ban singed/lee sin, but you may end up getting matched against much tougher competition after having waited 5-7 mins.
If you're looking for a way to ban in the meantime, however, I highly recommend trying ranked teams once you're level 30. My team and I have been doing it a lot and really enjoy it!