There have been a multitude of other reasons as well:
Raising the required number of champions to play draft (what level is reasonable if not 16, 20, 30, 120?)
The fact that champions aren't as broken OP or imbalanced as they once were, so bans are now for counters or preferences rather than banning those champions that are broken in the current patch.
That the draft mode process would take longer than it already does (it's already infuriating when you go through the whole process nearly maxing each pick time until the very end when someone dodges with 1 second left, and you just spent over 5 minutes for nothing).
And most importantly, as stated by Riot themselves (sorry, no link, this is paraphrased out of memory):
Increasing the number of bans doesn't solve any problems with the current bans. If currently you're annoyed that 6 champions are typically banned and the 7th-12th strongest champions dominate most games, increasing the number of bans will only change this so that 10/12/etc... champions are normally banned and the 11th-17th / 13th-19th / etc... champions dominate most games. It just changes the set of champions you see all the time, and has the added cost of more champions being regularly banned.