OK since URF mode was frowned upon by Riot and the community they decided to give ARURF mode, which is more cancerous since you get almost no say in your champ pick.
I have seen Sona already more times than I can remember, and been gifted Garen in almost 2 games in succession.
From what I can gather is that players prefer URF over ARURF, not necessarily because you can play the same free win champs game after game, but more so that you can actually choose a champion you know how to play because playing a random champ that is competently useless is more of a pain than losing to a Garen, Kat, Sona, ALI and Yi combo game after game.
At least when you got to choose your champion it was YOUR fault if you chose Udyr and not the enemies fault for choosing and face smashing you with a tanky Garen.
So, instead of just completely random, how about you give us X number of champs to choose from, say 5 each, and open the pool of champs to ALL champs, not just those you own.
Basically you give 5 champs each at random, that is 50 champs total out of 133, and allow us to choose one of those 5 with no re-roll options. Yeh you may get given 5 random champs you can't play and who are useless in URF, and may be given 5 who are OP and you just need to choose your best played champ, but at least you don't get a champ who is 100% useless and run out of re-rolls.