I don't think you're in the minority. I think WE're in the majority but why would we bother to come to the forums to comment that 3 bans per side is right? :P
I agree with everything you've said exactly - and more.
I like draft mode the most by far, and it was really hard getting in to it at first while trying to save IP for runes when I hit 20 and also wanting to buy "good" champions rather than just stacking the 450 IP ones. I ended up abandoning that strategy anyways because I figure I won't "main" 16 champions so I might as well just own some cheap ones, but at first it was important to save up and buy a 4800ip champion I would consistently use rather than a 450ip one I would abandon quickly.
Second, add in to that the fact that in solo-queue you're not strategically banning. Aside from banning "OP" champions or counters to the champions your early-pickers want, most people just ban "meta" champions. First, note the distinction between OP and Meta. Patch 4.21 Warwick was OP. Patch 5.2 Irelia is META. The point being, Irelia is definitely strong and has an advantage over other top laners because of it, but she isn't to the point that picking her allows you to single-handedly win the game against a better opponent. I don't like banning meta champions for the main reason that you don't know who mains them - the only meta champions I play are Ahri mid and Jinx ADC (although I just bought Jarvan 4 so him too soon), and in both cases I feel like the other team got incredibly lucky to ban them because they'd have been wasted bans if I wasn't in that game. Meanwhile, say we banned Irelia, Zed, and Rek'Sai against a team who mained none of those champions - the other team gained an advantage out of luck rather than out of any form of skill. Increasing the number of bans will only increase the occurrence of this luck. People will have their "main" banned more frequently and games will be unbalanced because they're playing at an MMR they earned with their main champions while playing a non-main champion.
As someone who plays a lot of Hecarim jungle, even before his recent buffs, and had absolutely no problem competing against the Lee Sin / Jarvan 4 / Pantheon / Fiddlesticks / Rengar / Kha'zix / etc... in season 4 and early season 5, I think I can safely say that meta champions are not OP. Meta champions are just popular because they're strong in many situations and thus work well in solo-queue where you can't control your allies picks. It doesn't make them overpowered in solo queue. I could agree with arguments that they're overpowered at the challenger level - where it's a necessity for early picks to take someone that can't be countered and that doesn't instantly give away the type of team-comp you're looking to build. Even then though, they're being picked for versatility in team-composition and not for power within the game itself. You shouldn't be banning a champion because it's versatile when picked but just normal once in game, you should be banning champions that are too strong in game.
So in essence, more bans are bad in my opinion. I'd only want more bans if things like 4.21 Warwick were more prevalent, but really I think in the 9 months I've played league I've seen maybe 3-4 occurrences of a champion being truly undeniably OP.