Play with friends or buy keys
Basically, riot made it almost impossible for players to unlock all their monthly chests playing alone. The key fragment drop rate gets extremely higher on a curve the more you obtain. Most solo players are forced to leave their boxes unopened, or buy keys. I get that they are trying to promote social play, but at least give solo players enough key fragments to open all of their boxes, and give premades extra keys for when they are too busy to play.