the class that has never really be done well.
the class that has never felt properly integrated.
Yes and yes.
The concept of the Monk was a very "of it's time" thing, in a way that no other class is, and it's never attained the sort of iconic distinction that, say, the Cleric has. I mean the Cleric was a weird-as-hell idea designed to counter Sir Fang, mashing up witch-hunters, the Bible, Bishop Odo and other sources, but D&D made it iconic, and now you'll find it or something like it in a million games, TT and CRPG (it hasn't quite every infiltrated literary or film fantasy, AFAIK, but that's another question).
Whereas Monk was maybe a little less weird, because it essentially had a single source - Shaolin Monks and the legends and myths surrounding them (though it picked up a few things along the way), but it was never as iconic, because that's so specific - it can't be generalized. It's not just say, "A supernatural martial artist"
Supernatural martial artist is what it should have been - that's the archetype, that's something that's appeared in mythology, fantasy, superhero comics and video games since basically forever. But instead of having an equivalent to the Fighter, Wizard, Thief, or the like, which encompassed these concepts, the D&D Monk is essentially a subset of "Supernatural martial artist", where they needed to create a class that was a superset, in the way that Fighter is a superset.
We should have had a class which Ryu, Kenshiro, Iron Fist, Scorpion, various Bruce Lee characters, Chun Li, and so on and so forth fitted into (maybe people Goku are bit too powerful for a D&D class, so not including him) and it can do some of those but not most of them. I don't really blame older D&D for not doing that - though it was a little weird they went with Monk specifically, but less weird if you think about the martial arts movies of the early '70s (often the hero wasn't a monk, but had been trained by them).
Where D&D has failed, is in failing to update that class in 3E, 4E, and 5E. By 3E, there was zero excuse for making a Monk class that didn't include people like Ryu, and which had such narrow and Shaolin-legend-focused sets of abilities - it was unimaginative, lazy design and mechanically weak. 4E was a little better, because you no longer had to focus purely on Shaolin-like abilities, but it retained the problem that the class, basically, should not be called Monk, or focused on weird supernatural monastic stuff. 5E basically reverted to 3E, which was pretty crap, frankly, and felt like it was a case of "We don't have any good ideas but we need to include this class". Mechanically it's a lot better than 3E, worse than 4E, but it's still called Monk, and still a subset of what it should be.
On top of all this, very concept doesn't even fit well into a lot of D&D settings, but the fact that it exists precludes a lot of other concepts from existing, because they cross over with it.
And this has massive knock-on effects on D&D so we get:
1) Attempts to make other classes good at martial arts via class abilities/subclasses
2) Attempts to make martial arts accessible via Feats, proficiencies, and so on
3) Attemtps to mimic supernatural martial arts stuff via subclasses, spells, etc. for non-Monks
And more. But because Monk has taken up so much ground, none of this ever really works well, at all. And yet MC'ing Monk doesn't work well, because of the way Monk stuff scales, and because Monk hard-dedicates with non-subclass abilities, tons of it's "balance value" to bizarrely specific Shaolin stuff.
TLDR the problem is that Monk is "Mythological Shaolin Monk" specifically and not "Supernatural Martial Artist". This was inexcusable in 3E and beyond. In 5E you can't even get around this properly by re-flavouring, because the mechanics, the hard-baked-in mechanics are specific to Shaolin-type Monks, not supernatural martial artists - this is why all the subclasses are all kind of half-arsed. The "basic" Monk should have been the subclass of a more general Supernatural Martial Artist class.