I think the new ban system is good and any problems will be fixed by themselves
While i understand that the new system might seem "random" since some bans can overlap i think that it is a necessary evil to let the system function properly - having to check every ban in a draft environment is really a pain timewise, and the longer the champion select goes the less fun the game becomes and the more are the chances that something may happen to the client or the player that denies the whole process ( like falling asleep half ban phase).
The new system simply requires to be adjusted to a new mentality - that you have not to ban the one you think is the strongest or more annoying for the team but the one that is for yourself. When the feature will be implemented we will see a lot of overlaps, but the more the people starts to think for themselves in ban phase the less frequent this will become. Also this makes the system itself effectively not "random" anymore, because it' s heavily reliant on your own skills.
Also cmon, having even two overlapping bans still means we get 8 bans > 6 bans. Having 5 bans completely overlapped is a real edge case scenario, and actually this makes me think how interesting would be to have a pick phase with a variable amount of bans.