"monk" means someone who studied a certain discipline to learn how to manipulate ki and unarmed combat, not "unarmed strike 1d4, +Wis to AC, and Con-save-or-stunned attack".
Ah,
that's where you've gone wrong, I get it now.
The fluff explains how you got your abilities, but there is no such thing as 'One True Fluff' for
any class.
What is 'true' is the crunch; the abilities you have.
Any fluff that believably explains those abilities is golden. How believable does it need to be? Well, at least as believable that attending a monastery enables you to run on water.
No fluff in the book represents the
only possible way to gain those exact abilities in a world with so many paths to magical effects. In such a world, it would be a mistake to imagine that Jason Bourne-type training undertaken by the elves of The Lachrymae Shevarash is entirely mundane and cannot weave the magic of the world into their techniques, to get the
results shown in the (Shadow) Monk class table.
If I say that their god, Shevarash, grants ki abilities, or that ki abilities just represent the natural magic that flows through the world and it can be harnessed by a disciplined mind, the idea that an elven mind can be disciplined through the particular brand of 'secret agent' training undertaken by some Lachrymae Shevarash is just as plausible as getting that kind of training in a monastery.
As DM, you can disallow any game mechanics that you want in your house. You may rule that there are no 'fighters' (in terms of the class) in your world. But this debate only makes sense if we talk about the PHB, not how you've altered it; that's not a common frame of reference.
But if you have allowed a class in your game, it's wrong to say that players are only allowed to use
your concept, not their own. You can make sure that they do have an explanation for those abilities, but the threshold should not be 'I wouldn't have thought of it that way', but 'Is that explanation at least as reasonable as any of the examples in the PHB'.
Some of the classes were inspired by real-world legends. Monks by eastern martial arts, paladins by the twelve peers of Charlemagne. But whatever the original inspiration,
we are not limited by those exact concepts.
Lets say you set a game in Charlemagne's world, but altered to include the PHB. The Twelve Peers, the Paladins, are an in-game noble private club. They were typical knights in shining armour...except the ones that weren't. Archbishop Turpin was a Paladin, but was actually the inspiration for the D&D cleric!
In such a game, having levels in the paladin class wouldn't mean you were a Paladin in game, and being a Paladin wouldn't mean you had any paladin class levels.
In 2E I made a paladin with the Savage kit, all without breaking any rules. She was a bit Tarzan-y (okay, a LOT Tarzan-y), but she gained her holy powers from her god (The White Jaguar, god of the natives of the island), disdained armour and used a spear. One comment I got (thankfully not from the DM) was "She's not a paladin; paladins wear heavy armour!" You make your paladins how you want, but there is no rule that paladins only get their powers while wearing heavy armour.
If I wanted to play a monk in the Charlemagne game, and the DM quite reasonably didn't want any eastern influence on this particular campaign, I'd have to come up with a plausible explanation of how I could do those things. TBH, I wouldn't want to, in a game which is all about knights in shining armour! But if someone else could come up with a plausible explanation and could fit it into the narrative, more power to him!
The DM could certainly say 'No monks....barbarians, non-human PCs...', whatever he wanted. But saying that there
are monks, but only if your character concept matches mine, is not the way forward.
If the campaign were based on ancient China, would you ban every class that was inspired by western concepts? No paladins, that's for certain, because paladins are only allowed in France, right? For me, there are plenty of Chinese character concepts that would allow a paladin. Every man and his dog can get abilities that leak into the supernatural, so a bit of Smiting powered by your Devotion...or Vengeance...no problem.
Oh, you want to play a paladin, but who only wears leather armour? LEAVE MY HOUSE AND NEVER RETURN!!! Everyone
knows that paladins wear heavy armour! It's my way or the highway. My players tell me they're okay with it...at least the ones who have no other game to go to...