Why not spice it a little more while we're at it, guys?
You ban out the enemy from playing - let's say Lee, but your own team can still pick him.
Also the potential mind games that this opens if they leave the teams seeing each other bans...
But that's actually less bans than the current system. Although all five bans would be effectively directed at the enemy team, that means you can only prevent the enemy team from playing 5 specific champions. As it is, with a ban restricting a champion for both teams, that's a total of 6 bans.
Yes, you get way more control over what champions are banned with your idea, because 3 of the bans in the current system are used by the enemy team; however, when there are a bunch of really high priority bans (when the number of ridiculously strong champions exceeds 5), you're leaving them at least one open, whereas in the old system, there was a better chance of getting all those champions banned.
Also, you take away what bit of strategy is left in the new system. With the new system, you can intentionally leave a strong champion unbanned, banking on the enemy team to use a ban on him, effectively letting you have an extra ban. This wouldn't be the case with your proposed idea; you'd just have five champions to ban and nothing more.
