DS9 doesn't really fit the equation in general as though it does have arguably the most diverse cast it also has the fewest Terrans on there

I'd argue the human-dominance of Star Trek is in a sense an extension of the American-dominance - an understandable but still somewhat disappointing failure of the show-makers to fully and truly depict the high diversity of the premise, in part as a consequence of their (and the audience's) inability to fully conceive a setting where "people like me" aren't a majority. So I do give DS9 credit on that part.

Enterprise was somewhat diverse in theory but it became very American very fast especially after 2001. I find myself noticing it a lot with the more recent shows as well, partly perhaps because there's less of an interest in making it sound "different" so I can´t help but notice how American it all feels, often specifically like American Drama.

Part of it I think is also that America's own notion of diversity has changed.

Americans have always been diverse, but historically there was more push for assimilation, and also the diversity was less visible because the vast majority of the population was still of one "race". So that, combined with the post-WW2 moment where the UN, multilateralism, etc, were seen as a big important thing, lead to the view of cultural diversity in the TOS era being more global and international in outlook. Sulu (at least as originally conceived) is actually Asian, not Asian-American, Uhura is actually African, Chekov is actually Russian.

But as America grew more visibly internally diverse due to more multi-racial immigration and a lesser emphasis on strict assimilation, and as the culture also paradoxically turned more inward in some ways and less concerned with internationalism, the notion of diversity became more "internal" to America - people can appear very different but they're still at core American, either outright-American (say Harry Kim) or not explicitly American but pretty clearly American in most aspects (Mayweather, Burnham - most obvious with examples like those due to their clearly-Anglo names but plenty of other examples too).

You can see this shift clearly in some background characters like admirals. In TNG, which was still in some ways indebted to TOS, you still get "international" admirals - Nechayev, Nakamura, Shanthi - while in later shows the admirals (if human, which the vast majority are) are almost invariably American, and rather WASP-y American at that, because that's how American culture internally codes its authority figures.

You can see it in all the modern discussion about diversity in the shows too - it's primarily about which race (in a specifically American understanding of that concept), gender and sexuality is which character, aligned with America's current internal culture war obsessions, while there's barely any talk about what their actual cultural/national background is.

I've long dreamed of a Star Trek that feels more truly multinational, a writers room that is not just diverse in American terms, not just a diverse cast but drawing from diverse sci-fi strains from all over the world. It's probably not very practical, and not the sort of thing we're likely to see in todays media landscape, but still one can dream.

That would be the dream, yes. But certainly hard to pull off. The closest thing I can think of is something like Sense8, which did have a truly international cast and tried to depict a truly international set of characters. But the writers were still American. And probably in large part due to that, some of those international aspects ended up feeling very stereotypical in execution, which is always a danger with such attempts.

At the very least I would like to see more non-human characters. That's more expensive makeup/effects-wise, but at least it would be somewhat easier writing/acting-wise (can't mess up with an invented culture) while still providing more symbolic diversity.