Lately I've become convinced that the root of the LFQW problem lies with the central mechanic of D&D levels: that a person can become infinitely more powerful just by practicing. This works just fine for casters, since the basic idea is someone who gains immense power through knowledge. A character knowing a thing implies a time when they didn't know it yet, so it's easy to picture a wizard at Level Elminster or whatever and work backwards to imagine his earlier, lower-level days.
Most other powerful fantasy figures don't get their power in the same way. Hercules didn't learn to be the son of Zeus; Nightcrawler didn't learn to be a mutant. Achilles gets dipped in the river Styx and Bruce Banner gets caught in a gamma bomb, but it happens all at once, not over the course of several years and a dozen levels. None of these characters really have a level one incarnation. A suitable martial archetype for dungeons and dragons doesn't just need to be Hercules. It needs to be a nobody who can pick up a sword and become Hercules one day just by working really hard, and that's not something you see in Western fiction very often.
That is why, fiction wise, I've been playing with explicit power sources as mechanical hooks.
Mechanically, a power source starts out as something like a feat or an attuned magic item (a small to medium sized thing attached to a character), but you can use it to justify
gaining levels in a class. And as you do so, the power source also gains power (becoming like a feat chain, or a more powerful version of the magic item).
Power sources might have a max on how many levels they can be used to justify. And they can be found in-game. And all PCs will start with one (which also explains why, mechanically, you gain competence so much faster than random people do; the world simulation problem).
Your power source could be
Divine Blood (1) -- which starts out giving you death ward once per long rest. And it might be good for 5 levels (to start!), upgrading the death ward to 1/short rest, and maybe another feat-like ability you can pick from a menu.
Maybe later you kill a dragon, and bathe in its heart blood as you slay it -- and gain the
Dragon Blood power source. Which gives you fire resistance initially, but could go a lot of directions from there (scales? fire breath? Turning into a half-dragon? Lots of ways!)
Like magic items, you can have a limit to how many power sources you are attuned to.
Anyhow, once you have these power sources as justification, martial abilities at higher levels can be insane without asking "why can a street urchin do that?". You can teleport as a high level Rogue? Well, you do have divine blood! Or, you bathed in the blood of a dragon. Or you swallowed the heart of the time lord. Or you practiced using ancient, mystic scrolls. Or one of many, many power sources, all of which justify "you are awesome" narratively.
This also works for spellcasters. It should take more than a few weeks to become an archmage -- but not if you have the soul of a demon prince, the blessing of a fey lord, and have read the entire endless book of all that which we do not know.