I don't believe the issue lies with how many bans we get or how many "broken/problematic" champions there are. I believe the issue is that no one (at least, from my Silver Rank point of view) understands what the bans are truly meant to be for. The bans are meant to be to ban out any champion that could possibly counter the team comp you are going for. The problem with that, though, is that (again, from my Silver Rank point of view; notice the "Silver") no one picks their champ(s) based off of a possible team comp that they could be using. Which is usually why players ban out the FoTM champ instead. I, too, am guilty of this and that's solely because I don't have enough experience in the game to know what champs specifically counter what comps.
Now I have heard that as you climb up the Rank ladder, champ select becomes more strategical, with bans being based off of team comps and team comp counters. This is how the bans should work in all elo's, though, imho. You want to be a poke comp? Then ban healers since they practically negate any/all poke. Want to be a siege comp? Then ban champs with lots of waveclear. Want to be an engage comp? Then ban disengage champs. Want to be a splitpush comp? Then ban champs with heavy engage. If more players from more elo's (specifically low Gold and below) were to use comps and bans like this, I would honestly be able to say that more bans aren't truly necessary.
But, again, I'm just a lowly Silver III player. What do I know?