I think the big problem with high-level play is that it tends to be fairly boring for various reasons.
1) Casters start being able to do ridiculous things to the point where it can be increasingly hard for a DM to even think of solutions. Banning a bunch of spells can fix this, but it's certainly a thing and even for a lot of caster players it's not very interesting. This is a fundamental design flaw of 1/2/3/5E D&D (not 4E, which had different flaws). Casters can simply do too much, too easily, at too little real cost to themselves, outside of combat (inside they're more balanced). The higher level you get, the worse the issue becomes.
2) The monsters get more complicated and overpowered and this also tends not to make things more fun for a lot of groups. Especially when you need to start including multiple complicated and powerful-seeming monsters in the same encounter. It's easy to overdo it with fancy monsters too, tonally.
3) HP get so bloated every starts seeming a little silly. The math still works in 5E, but the numbers are kind of videogame-y in a bad way.
4) Proliferation of magic items often reaches excessive levels. This can be mitigated, but it can very easily add to the complication and boredom.
In general I don't think many would dispute that D&D tends to peak in fun somewhere between about 3rd level and 12th level, and these are big parts of it. Outside of 4E, D&D has never had a particularly good framework for high-level play (and again 4E had other issues). That's still true in 5E, just the math works better in 5E.
One thing is very obvious from this thread too is how little and low quality the guidance and support WotC offer for higher-level play.
Also a fundamental issue is that an awful lot of people see going up the levels as some sort of test of worth or skill, and creating high-level characters as wrong or cheating. That common attitude obviously severely limits how much play high-level stuff will see.