What ARURF is telling us
"We don't care about you enough to try balancing URF."
There has been little to no feedback that says ARURF is better compared to picking our own champions. All that making it random does is ensure that when the enemy gets Ezreal, Hecarim, Alistar, etc., there's literally a 90% chance we got someone weaker.
So here's a few suggestions for bringing back the mode we actually liked, while keeping it balanced:
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10 bans is the most obvious one, and should've already been done. The pool of champions that are too overpowered in URF isn't too much bigger than that, and a few of the offenders have been changed anyway.
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Set a strict limit wherein players are not allowed to play the same champion twice in a row, but alongside the 10 bans.
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Like you did for Kassadin, remove any additional benefits a champion might get from spamming a single spell. Take away Ezreal's passive, remove the CDR from Hec's Q, etc.
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Halve the duration of all knockups since the bonus tenacity from the URF buff doesn't affect them. Since I'm sure recoding tenacity so that it does would be too much to ask.