This is where Fate (among others) appeals to me, as well. 5E D&D has a "sweet spot" of 5th-10th levels (or so). I can deal with the 1-3 "training" levels, but have been starting games at 3rd. I've also made it clear to my players that things will wrap right around the time they get 5th level spells -- and it's really the spells that bug me, though oodles of HP aren't a winner, either.
This matches my experience as well, although plenty of things in D&D beyond the spells manage to bug me as well.. It seems like once you get to about 10th, there's just too much to keep track of and the story just drags and drags behind the weight of the combat. Plus all the attendant homework for the DM. I just ran a 5e game up to level 8/9 and was
relieved when the party TPK'd on a trap.
I'm not totally sold on just how little advancement there is to Fate. I'd like to see the characters grow some. But, I haven't played in a campaign of it, yet, and it's easier to add in rewards than it is to take them out.
I've run a few Fate campaigns, and I've seen two issues with advancement in "default" Fate:
a) Fate presumes that characters are pretty competent right out of the gate. You don't have to wait through two years of play to gather enough feats and skill points to have the character you envision. However, that makes it harder to pull off the zero-to-hero thing with strictly the default mechanics. You can pull it off more easily with changing aspects to reflect the character's growth, or slightly less easily by tweaking the character generation/starting point or Stress etc.
b) A "+1" in Fate is kind of a big deal, compared to D&D. This makes the advancement part of growth especially tricky in implementations like Fate Accelerated, with fewer places to put them. (It also leads to the funky rules about which skills you can advance and when to keep your skills in a "pyramid".) Personally, I just create a list of things like "Raise a skill from +1 to +2" or "choose a stunt", and you get to pick one each time you reach major milestone (different in Fate than D&D), but only once.
IME, these aren't really all that big of a deal, except for old grumpy D&D players who really expect a traditional "leveling up" experience. Since Fate can efficiently cover more plot/story, the focus on that generally makes up for it. The real trick is realizing that Fate isn't about your character "leveling up" and is instead more about your character's growth. (Unless, of course, you have
Took a level of Badass as an aspect.)