It has been suggested, but either they are working on it and haven't delivered yet, or the right people haven't noticed the idea yet.
I would love it if they added these, particularly for bans, but it would be nice for picks too. Think about it like this. If you have a tab, you can cycle through selections that are good for the team composition and fit your style much easier since they could be all on the same row instead of scattered throughout your champion pool. While if you use the default tabs, they won't even be on the same page. You could also cycle through champions you don't really care about but are popular(or unpopular if not duplicating is your strategy) bans that you suspect your allies would want you to ban. And they could ask you to ban one of them. As it is, I just ban no one because the entire champion pool page is too cumbersome to cycle through the various ban propositions.