Possible URF compromise

The game mode remains random, with the full champ pool unlocked (or owned+free champions unlocked, either or). You simultaneously accumulate rerolls, as well as a much more slowly accumulating free pick. After so many games playing in URF, you would accumulate a free pick in which you could pick a champion that hasn't already been picked on your team.

I think this would reduce a lot of grievances with URF. The games would still be largely random so that you won't see Zed, Kata, Veigar, etc every game, but if you really wanted to land a champion and feel like the RNGesus isn't giving you that sweet, sweet sweet full attack speed jhin build champ you want to play, you can occasionally pick them regardless and get the experience that you want with that champion.