The best way to implement 10 bans.
Adding more bans to the current system and more time is not the answer.
Let all bans happen simultaneously for both teams. There are two clear options for this.
Either both teams have a preset time, let's say 60 seconds, to ban. They can see who the other team is banning and react accordingly. You can choose to ban someone who counters your pick or the most imba champion out there. If you wait too long and don't lock in your pick you simply don't ban anyone. The game doesn't remake. The game goes on with less than 10 bans. If no one bans anyone then all champs are still available. This allows for some mind games, doesn't add more time, and doesn't kick people out of the game for missing their ban.
The other option is to have both teams blindly ban five champs. Same basic 60 seconds time limit. They cannot see the other team. So if both teams ban the exact 5 champs then only five champs are banned. If 10 different champs are banned all are banned. Like in the other example if you don't ban anyone you don't get kicked out for not banning a champ, you just don't use your ban.
Both of these speed up the process and allow for plenty of bans.
Edit: for solo/flex queue only. Not for professional play.