All Random URF is a great idea, but...
You're on the right path. URF is much more enjoyable with everyone having random champions. However the problem is, there are tons and tons and tons and tons of people who just leave queue and don't care if they have to wait 15-30 minutes afterwards.
So, this is my suggestion to help lessen this.
If you're going to give us random champions, give us everything random. Pure random. Like, we don't know what we have. At all. Random Summoners, random champions. Then we'll find out what we have 'when the game starts.' And I mean like, we won't find out till the champion load screen.
Is this a perfect plan? No. But it would certainly help with lessening the people who leave in champ select.
If they wanna leave during the game? They risk getting banned. And that risk is a lot more potent than 20 minutes of wait time.
URF is supposed to be pure crazy. Absolute insanity. Why try to make sense of it?