It’s interesting, isn’t it. We don’t hear Mongoloid or Negroid in the US. That isn’t because the US has suddenly turned away from Racial categories; they call them Asian and Black or African-American. And add to that Latino (very much an American classification), Native American, and Islander.
But Whites have retained Caucasian in the States.
Of course, one of the salient things about racial categorisation in the US, which is alien to the Old World, is that the most salient social tribal groupings within the US have not been based on ethnicity (whatever that may mean), as is familiar in the Old World. Whites as a tribal grouping in the States results from a melting pot of European cultures; English was the common language, but people didn’t rally around an English or even British culture—too many Irish and Germans from the very start. The same of course goes for Blacks in the US, with the added component of repression and cultural suppression.
So whiteness is a big deal in the States that it wasn’t elsewhere. Other whites, even slave-owning whites, would identify by ethnicity first and skin colour second; that wasn’t quite an option in the US. There was of course the new identity of American; the problem with that was, it didn’t adequately exclude blacks.
So. That’s why whites get referenced much more in the US than elsewhere. But why Caucasians and not Negroids?
My surmise (since I don’t get good guidance from Wikipedia this time) is that in the late 19th century, White Americans were aware that their allegiance to race rather than ethnicity was an odd thing by European standards, and were eager to find scientific backing for it. “White” and “Black”, and even more now “Yellow” and “Red”, sound like a bizarre focus of allegiance: you’re proclaiming allegiance to a pigment, not a tradition or a community. (White people in Europe sure didn’t feel they all belonged to the same thing.) Racial Anthropology supplied that for White Americans.
White Americans did not feel the need to speak of Negroids or Mongoloids outside the context of anthropology or eugenics. I think that’s because the scientific discourse was more about ennobling White Americans’ self-identity, than about systematising their notions of other races. Caucasians may have felt funny paying allegiance to a pigment; they didn’t feel embarrassment from reducing other races to a pigment.
Yellow and Red are long dead. The opposition of African-American to Black and Negro came from the astute observation that Blacks had an ethnic identity in the US, comparable to all the ethnic identities of “hyphenated Americans”, which could be opposed to a pigment. That move has its own problems: South African Americans and North African Americans are not who it is meant to designate; and “African” is an ethnic identity only in America, for the same reason “Caucasian” is an ethnic identity only in America. But it has been successful.
And that leaves Whites. Caucasian was appropriated from the framework of Racial Anthropology. But it was recontextualised, and it was never thought of as derogatory. You can argue that Negroid and Mongoloid are derogatory; in fact, it’s hard to use those terms now with a straight face. But Caucasian is no longer defined in the same paradigm as Negroid and Mongoloid: Americans have forgotten all about the original Racial Anthropology framework, and I doubt most users of “Caucasian” have even heard of “Mongoloid”. Caucasian is now defined in opposition to Asian and African-American. Its etymology is now just an historical accident.
So in short: the legacy you allude to for Caucasian, OP, has been forgotten about. The word is being used in a paradigm that owes a lot to that legacy—but that existed before it, and have survived after it.