"Bad communicators"
What if, say, there was a report option: "bad communicator" - that players could use to mark someone as a bad communicator, which would prevent them from being matched with that player for some amount of time (say, 7 days). This would vary in impact depending on how much both players play at the same times - ranging from none to very little. And this is good - because it could've been a fluke - in which case there's very little impact on overall matchmaking times.
But say this player is a habitual bad communicator - doesn't ping well, spams chat with "toxicity" or doesn't help make calls, etc. - this player would be reported as a bad communicator by many more players over more games - greatly increasing this player's queue time and having a not very big impact on other players (it'd just be as if that player wasn't logged in, really). Once someone got enough reports for them to start noticing a difference, Riot would pop up a notice explaining the increased queue times, and with advice on how to communicate better to avoid reports for it from others in the future.
The good news? This player reads that and improves - they ping better, they discuss and shot call in chat, etc. So they stop getting reported. Since the matchmaking penalty only lasts at most 1 week, it takes this player no more than 1 week after they stop getting reported for it to return to normal matchmaking times. And in the meantime, again, it doesn't affect other players at all - it's just as if the player wasn't logged in (the matchmaking pool is 1 smaller). In the grand scheme of the tons of people logged in, one doesn't make a difference. Even a few wouldn't make a difference.
If the player continues being a bad communicator, they would continue being reported by people, and they'd still keep large matchmaking times for themselves.
I think this idea provides both a solution to players who want an "impact" in "self-policing" like Tribunal where they could "vote" on what is toxic behavior or not, but also a solution to players who complain about lack of communication. This would allow the playerbase as a whole to determine what is good communication or not. Through this temporary divide, the playerbase will set standards for behavior in that players who are getting reported will improve their communication until it isn't reported anymore.
To help prevent abuse by false reports, this feature could be enabled only when 2 or more non-premade teammates report for this maybe?