More bans of course would do that, that's not the argument against the increasing of bans.

I'm against increasing bans, so are a lot of players. It has nothing to do with your point... but I'll give you my opinion why I don't like the idea.

For one, I play soloq for fun and to practice getting better. I personally never put much time or effort into the bans, I'd prefer learning all the MU's properly and being ready for anything. The only time this isn't true is during certain patches, when some specific heroes are super over the top and I know they will be nerfed. In this situation, I ban them. As I recall, there has never been over 6 of these, and in most cases it's a single champion. In no super-specific order...

Since last year(that I can remember...) Rammus late S4. Warwick late S4. Reksai on release. Skarner this season. Sejuani this season. Azir. That dumb Nunu Q buff where he could out smite anyone level 1...

Sometimes there's multiple champs at once that get strong because of an item, but realistically bans usually don't do much to stop this. For instance, the Cinderhulk meta... Now in this situation, more bans would kind of help I guess, but I mean the cinder hulk meta for all purposes was half of Season 5. It was something people basically hard to actually start to learn and play to succeed. I'd argue banning 10 tanks doesn't really fix the problem with bad item design at all. Once you have that many champs that are a "problem" it's simply the meta. An example of this is the Marksmen role. If there was only one marksmen, it could be a problem. If there's 5-6, it's bannable. If there's 20 of them... it's the meta. Ban 10 of them if you want but there's tons left.

Usually the jungle item changes cause the most amount of complaints about bans. In fact, literally every super OP champion in the last year and a half that I remember has been a jungler, half of them because of jungle item changes.

Basically, it's really rare to have one specific champion who's beyond broken and 100% ban worthy. Usually if there's a ton of champions you want to ban at once, it's because Riot released an OP release item(Feral, devourer, cinder, and so on.)

There's a few exceptions since even S3... and it has been champs like Kass, Lee, TF, and so on. Lately, most "OP" champs are related to patches, and they get nerfed within 2 weeks. There's usually at most one or two of these.