One thing I just realized would be necessary for this to work is the ability to choose which chests we can get rid of when we win more times in the tower than the slots we have, instead of having to get rid of the chest we just won because we don't have enough gems to open it. That way, when you have a bunch of common chests, and you earn an epic chest, you can get rid of one of the common chests, and place the epic chest there, so you don't have to get rid of it if you don't have enough gems to open it.
It might even be easier if the chest quests are completely replaced with, 'win so many times in a row in the tower,' so that kind of change would not need to be made.