Honestly I have been reading these boards for awhile now and haven't been able to find many posts about increasing the ban number.
They pop up every month or so. I think I've seen around a dozen since I started frequenting the boards.
How can you possibly call yourself a pro if you can only play 2-3 champions exclusively? Coming from a Diamond player I find this absolutely inexcusable.
It doesn't matter how many champions you can play if there are only 2-3 good ones. Amazing can play Volibear and Xin Zhao, no question. He's going to get roflstomped if he does it against a pre-nerf Evelynn, tho.
Pick and ban is a lot less forgiving in competitive play than it is in soloq - an advantageous jungle matchup might not be the end of the world in Diamond, but if the jungler with the advantage is Dandy or KaKAO, you're gonna get fucked pretty hard.
I see a lot of people arguing that more bans would force players to have wider and more versatile champion pools. I disagree - I think that it would have a homogenizing effect, and here's why:
Let's say that you're a jungler playing in a pretty bad meta for junglers - Evelynn, Elise, Lee Sin and Kha'Zix are all pretty OP, while Vi and Jarvan are semi-viable picks. If there is a single champion in that list that you cannot play, and the enemy team has four bans, they can ban any four of those champions and first pick whichever of the remaining two you can play. The only way you can prevent that from happening is by spending at least two of your own bans on junglers. And in so doing, have only two bans total for the OPs, and be forced to hand over the 3rd strongest champion in the game to your opponent.
As you can see, as bans increase, the risk associated with not playing even a single meta champion does as well. With three bans, a jungler in that meta could comfortably not play one of the perceived OPs without putting his team at risk - with four bans, that jungler had better learn Kha'Zix whether he likes it or not.
Counter intuitively, the fewer bans there are, the more freedom pros have to differentiate their champion pools from one another. Which I see as a very compelling reason to avoid increasing bans in competitive play.