What I hate about 5th Edition is class design from 8th level and beyond, especially for Fighter and Barbarian. Other than subclass features, which for a Barbarian end at 14th level and whose subclass captones really feel mediocre most of the time, most classes (other than casters w/ higher level spells) get absolutely nothing fun at these levels. Indominable is not a fun feature. Brutal Critical could be, if Barbarian also got a bonus to its critical hit range. And most subclass features for any class, with a few exceptions (Warlock), are super boring at the top.
I think the main reason play above 12th level doesn't happen, and especially past 14th level, is because a lot of classes get a lot more boring at those levels. The "story" of your class as a fighter is no longer progressing. Hell, even your story as a Cleric is no longer progressing, or any of these classes; they put a bunch of bland features up there that just don't make those levels engaging to narrative players.
Whenever I make a homebrew class or any supplementary game material for higher levels, this is always on my mind. A Fighter above 10th level feels the same narratively as one at 6th, 7th, 8th levels. A Barbarian too. Instead, they should be climbing to paragon status, which there are literally countless ways to represent mechanically. Everything from strongholds and followers, to superhuman feats, to hyper-effective combat arts, rallying the troops, and so on.
Also, I don't like the community's love for hardcore balance when 5E is not a hardcore balanced game, only a moderately balanced game. For example, at level 1, the Paladin starts with Lay on Hands, the Fighter with Second Wind. Lay on Hands is a better Second Wind in every way, and scales far better into the game. Everything from healing, to removal of diseases and poisons, and that compares to Second Wind how? I understand not wanting another 3/3.5E with crazy balance problems, but I don't think its sensible to pretend like 5E is an iron-clad and even game. It isn't.