I DM'd 5e for about 4 years before I got tired of it. It's a great system. It shits all over 3e and (heresy alert) AD&D 2e. I eventually got tired of it for a few reasons though.
Before I tell you those, here are my three favorite things about it:
Everything is unified and simplified. It's way streamlined compared to pretty much any other edition.
Advantage/Disadvantage is a wonderful mechanic that you, dear reader, should steal for your own 'old-school' game. It makes just the right amount of player-DM negotiations/lobbying happen, gets rid of a lot of troublesome ambiguity, and saves a lot of thinking about how many bonuses a player should get for trying some clever action. This was probably the best addition to the system.
Rogues/thieves are actually fun to play. In old-school games, thieves are unbelievably boring and needlessly complex. Their skill trees make no goddamn sense. Lamentations of the Flame Princess has an interesting solution for this, but none of the other systems do. Even in my personal favorite system, DCC, thieves are stupid as fuck. When I DM DCC, I allow my players to take a 'rogue' class that is basically a 5e rogue hacked into DCC. 5e does this right straight out of the box.
I eventually left 5e behind so that I could use the many modules reviewed on this website with no fuss. With more experience DMing different systems, here are the problems I think 5e has:
Combat takes fucking forever. Holy shit it is slow. When I started DMing DCC and 1e I couldn't believe how fast the combat went. What would take an hour in 5e takes 20 minutes in any B/X knock-off.
The rest/healing system is really hard to work around if you are trying to incorporate resource management into your game. It's one of the least modular parts of the game, and it's hard to just shave off. I tried using several variations of the rules but none of them felt satisfactory.
Everybody has fucking dark vision arrrrg I want darkness to be scary not comfortable.
Characters are way too resilient. As you level up, it becomes basically impossible to die.
Those are the biggest problems with the system, which all and all, isn't very many problems.
I agree with DP that a big problem is the material being released for it, but I think the biggest problem is actually not that.
It's what everyone has kind of been hinting at. The problem is player/DM culture. People have super weird expectations about their characters and how the game should be run. It's almost expected that you show up with an 8 page backstory and a progression for your character from level 1-20. Your character is supposed to last you for years, and if they die, it should be an epic story moment. All of the 5e games I have played in are like playing a video game on easy-mode. The dungeons are designed to be cleared out in one run. There is very little jeopardy and the DMs are expected to go out of their way to soften punches. They roll behind screens for everything and make the monsters do weird unreasonable things to stop characters from dying/stop the story from being derailed.
This isn't 5e's fault so much. Maybe it is Wizards fault, but I think it is more just the zeitgeist. If you take people out of their 5e/3e expectations, and run an alternative system, you can run much more interesting/challenging/deadly games with no push back because nobody can show up with their stupid 8 page back-story.