White has become a "term" that reflects that the degree of success of social and biological integration between European ethnicities. As mentioned, people did not refer to themselves as white when I was young, but today, very few American born people are any one ethnicity and so ethnic identity is being lost.
My parents belonged to a German club. And I have visited Italian clubs and other ethnic clubs that existed when I was young and some still remain.
But for the most part, American citizens of European origins no longer identify with their European ethnicity and so accept the "label" of white (even as many resent the label), much as many African Americans resented being referred to as "black".
It is not unlike African Americans, most first and some second-generation blacks identify based on their tribe or nation in Africa, Haitian, Jamaican, etc. But after two to three generations, they identify as African American. The first stage of social and biological integration. So, my daughter-in-law identified as Shona, not Zimbabwean and my son-in-law identified as Nigerian not African, and my granddaughter's father identifies as Haitian, not African American. You might call them African American or black, but they don't call themselves by those labels.
Ditto, for Hispanics and Native Americans. I have friends who are first generation Cubans and identify as such. I have friends that identify based on their tribe. But most of my Hispanic and Native American friends are increasingly identify as Hispanic and Native American, not their ethnicity or tribe.
My children and grandchildren are going through the next stage of integration from ethnic integration to racial integration. Mixed race is the fastest growing "racial identity" group in America measured in percentage terms.
People like my grandchildren reject being called "African Americans" (even though they are more "African" than most, since their mother was born and raised in Africa). Instead, they want to be called Americans and when asked to identify racially by government documents indicate "mixed race".
My two youngest daughters were very confused as children. One said she was Japanese because Japan is halfway between America and Indonesia when you fly Japan Airlines between the two countries (which she did often as a child).
Another declared she was Hawaiian, when we went to Hawaii because everyone there looked like her. Today, both girls identify like as Americans of mixed race. And dislike being identified as either "white" or "Asian". Skin color or genes do not define them, so why should they be labeled that way.
This is not unique to America. My nephews and nieces on my wife's side of the family identify as mixed ethnic Indonesians (the equivalent of "white" in America today), their father is Chinese and their mother is native Indonesian (a mix of two Indonesian ethnicities).
Regarding "white" is has been confused and ill-defined for decades. What is "white" based on... not skin color, there are "whites" who have darker skin than blacks. Also, your statement that "white" was never used amongst people of European descent is absolutely wrong.
For decades, northern Europeans referred to southern Europeans as "not white" including Spanish, Italian, Greeks, and "Roma". Today, all those groups are now included under the "white" label used by our government tracking racial data.
Even today, the term "white" has been wrongly used to differentiate European whites from Hispanics, Middle Easterners and Indian "whites". Now the census, breaks out "Whites" as non-Hispanic whites (whatever that means) but for what purpose. In California, non-Hispanic whites are the second largest "racial category" (but how can you have a racial category that ignores over 30% of the people that are racially white). Moreover, one third of the remaining whites (excluding the Hispanic label) are not European whites. So, what purpose does the label serve?
Interestingly, while "genetically" Indians are "white" using today's "racial" definitions, Indians often to identify as African American or black, while Sir Lankans and Pakistanis tend to identify as white on surveys.
Who knows how many blacks are actually "white"? As a sociology professor I know has said, race is increasingly a "self-defined" term. Even as the self-appointed "race police" try to prevent such behaviors.
The whole labeling process makes no sense as nations going through the integration process lose those distinctions over time. We should end it. However, it provides politicians a great way to divide people and so it is likely to persist.
I do find your comment regarding "white" as a socio-political construct since in New England, Jews and Asians, who have historically been considered "minorities" are now excluded from those definitions for many measurements. As an example, Asians are no longer included in Massachusetts when measuring what percentage of children are "minorities". Jews have been defined by the "woke" movement as "oppressors" even though they experience the greatest amount of "hate crimes".
So, your point that "White" has lost its "racial" connotation is correct, instead it has become another term for any group that is defined as white in the new "woke" culture wars.