Suggestion: Only Allow Champions' Ranked Use at Tier 5 Mastery
EDIT: Due to how recent Champion Mastery is as a feature, this might not be something to implement immediately; rather, I think this could work after everyone has had sufficient time to show experience with champions they've already mastered to a degree, so likely a season or two.
Right now, I feel there are two separate problems with Champion Mastery and Ranked play. On one hand, Champion Mastery, while a cool way of earning bragging rights, doesn't feel too useful or relevant in the long run, particularly since it carries very little reward beyond just another badge. On the other, Ranked mode is unstable for a ton of different reasons, especially following champion releases or major reworks, partly because a lot of players take flavor-of-the-month champions, or simply champions that seem interesting, and test them out in a competitive setting with little to no practice. I feel a solution to these issues would be to connect the two systems: disable all champions for Ranked play as a baseline, and have tier 5 "unlock" the relevant champion for use in Ranked mode. Ranked mode itself could then be locked until players have a certain amount of "unlocked" champions (say, 15).
The major pro to this would be that it would tighten the entry-level skill requirement for Ranked play without affecting anything at higher levels: serious Ranked players have almost definitely mastered more than the required amount of champions, and players who have an insufficient level of champion mastery should probably not be diving immediately into Ranked mode anyways. It would also add a feeling of reward to Tier 5 mastery beyond just a shiny medal, and could encourage players to try out those champions in Ranked in a healthier way ("I have a fair amount of experience with this champion, maybe I should try them in Ranked", rather than "this champion looks cool/overpowered, maybe I should try them out in Ranked mode and see what happens"). A con to this is that it would significantly delay entry into Ranked, though I feel that would be a good thing, as Ranked mode is meant to be about high-quality play, which isn't achievable right off the bat.