But there would have practically no ranked integrity unless the random aspect was removed.
That's not necessarily true. For instance, card games (e.g. poker, mtg) have randomness as a fundamental aspect, but they're still generally regarded as competitive. Of course, part of that's due to playing more games to balance out the randomness. However, the ranked splits being introduced next season encourage players to play more games anyway.
There's also ways to restrict ARAM variability without outright removing RNG. Bans would be one option. An "arena draft" system, where players choose their champ from a handful of random offerings, would be another option. Distributing champs based on relative power levels would be yet another option.
People already think twisted treeline ranked is a joke when it has champs with 60% winrates and some down in the 30s
This is tangential to the larger point, but you massively oversimplified things here. The actual state of Treeline balance is only (a small) part of the overall League playerbase's feelings towards TT, considering most players have barely any notion of TT gameplay. In general, Riot's prolonged disregard for the map has made it annoyingly common to think that the entire map "is a joke." Plus, 3v3 Flex has a similar connotation as 5v5 Flex, which isn't well-regarded either.