There's 3 main issues with more bans:
- champion pools of people playing the game
- extra time
- Wanting more bans to fix Riot's balance
Champion pools have always been fairly small in LoL. While you can always point to teams like SSW or a star player on each pro team as having a big pool - those aren't average players. Those aren't even average pro players - many of which only have ~3 champions they're at a pro level on.
And then there's you're average player - who needs to grind for months to get the 10 necessary champions to even play ranked. If you want to own enough champions to actually play any roll without getting banned out with 3 champions - there's a decent investment in IP, especially if you find you're not really into champions you thought you'd like after 10+ games. Bump the number up to 5 per team - and you can easily and regularly ban out an average player by accident, let alone LoL King them and do it on purpose.
For extra time - more bans will take a little more time to punch in.
Average game length is around ~33 minutes from what I've been able to find about LoL, and most TV sitcoms are 21 minutes with 9 minutes of commercials. You probably don't want to hit that ratio, and you certainly don't want to go over it.
More an issue of an extra 30 seconds not being a huge deal, but a lot of tiny additions of 30 seconds without considering the added time can end up being a huge deal - so it's something worth considering.
Then we can perma-ban all the broken champs until Riot finally fix them
This is the biggest reason not to add more bans though.
It gives a really awkward message to Riot's balance team. It's ok to make something crazy OP to test out because it will get banned?
Also gives a weird message to the community - Riot's balance team has thrown in the towel and it's up to the players now?
Not to mention more bans doesn't actually solve any issues in regards to broken/OP champions. Every data-tracking site that looks at LoL champions doesn't have ~15 outliers at the top. Champion balance is just one fuzzy, gradated line that pushes some from ~45% to ~55% win rates. Even if you gave 30 bans per team you'd still have to many "broken/OP" champions to ban out.
Broken/OP is more of a psychological, player-based issue for LoL than an actual balance issue from everything I've seen. The meta favoring ~3 champions in each given lane position strikes me as having everything to do with gold, silver, and bronze medals being a big deal in western culture and people being to lazy to remember more than 3 things.
As such - more bans is a band-aid solution to such an issue, if it's even a solution at all.
Obvious pros of more bans is more greater tactical leverage for banning out specific strategies and play styles, but I'm not sure that really outweighs the cons.