There is a saying in Magic: The Gathering:
"Free information is free."
In that game, it means that pro players never give away any free information about their hand, board state, or intended actions, because this free information helps their opponent choose better and more efficient responses against them.
Translated over to League, under your proposed system, the team that bans first is giving free information to their opponents. Said opponents can then use that information to make better choices and counterbans and counterpicks.
In other words, if your opponent bans first, then you get to see your opponent's bans before you do, and these bans may reveal things to you about their potential positions and playstyles that you can use to make more efficient choices. As most people playing this game for any length of time have found out, very skillful players sometimes only need a slight edge in efficiency to begin a snowball.
TL:DR - Whoever bans last gains an advantage under this proposed system.
In high-level play this naturally must lead to a lot of people delaying their bans as long as possible to avoid giving away this advantage. I think we'd probably see a lot more accidentally dodged matches at higher ELO as people try to time their bans to the end of the timer and miss, and lower-skill players being somewhat taken advantage of in lower ELO due to not understanding what free information they give away, if this system were implemented.