Let's utilzie champion mastery for a more useful purpose!
A lot of grief in rank stems from players playing a champion that they have minimal experience with. A second grief is when you have smurf accounts come up who play one champion from 0-30 and then smash low level leagues. No one likes to play against a diamond shaco in bronze/silver.
One way to prevent both of these problems is to have a minimum champion mastery rank before playing a champion. For example rank 3.
This makes sure that players who decide to play a champion in a ranked game are at least familiar with that champion, and I think that this makes it fair to the rest of their teammates. If you want to learn a champion do so in a normal game.
This doesn't complete solve the smurf accounts, however it does make sure that the smurf account has played with a variety of champions in normal instead of stomping with one or two champions, making it more of a pain for them.
The downside of this idea is that it would require players to level multiple champions up to rank 3 to ensure that their champion pool is large enough so that they can actually queue for rank. However if the community is given notice ahead of time of the change, people could easily diversify their play in the current ranked/normal games to get the masteries, and business goes on uninterrupted.