We don't get bans because the mode is random. You can get bans or randomness, but not both.
The randomness is designed to give all Champions relatively the same amount of play time (though dodges, rerolls, and the popularity of Champions in players' Champion pools will make the same playrate for all Champions impossible).
With bans, the best Champions don't get to see play time, but then the next best Champions get the spotlight. It doesn't really solve anything and still makes the meta stale. You're going to get much less of those same picks in random URF.
I realize it's not fun to play against strong Champions, but you will always see strong Champions if you have bans or randomness. With randomness, you'll see less, because people won't be actively picking the strongest Champions.