Not the american mindset, the human mindset. Humans will always find differences between groups and go to war over them. Even in african countries, where everyone is the same to our eyes, neighboring villages will fight eachother over perceived differences. It's just a basic part of human nature. Or look at Japan before the black ships arrived. Fact is humans need enemies to fight, and we will always find them.
My point is that groups do not oppress/discriminate/fight other groups because of biological differences, or even cultural differences (religious, linguistic, dress, etc.). The motivation is always economic or political. Those differences are just convenient criteria that the upper group adopts to draw a sharp line between "us" and "them". Biological differences, when they exist, are "better" for that purpose than cultural ones, because "they" may change their customs and religion, and quickly learn "our" language; but "they" cannot change their skin color, and their children will inherit it.
That is also the reason why racists do not recognize gradations, or put all "mixed race" as a single separate race. Social discrimination is by necessity a binary thing: either "they" are allowed to attend medical school, own land, live in this neighborhood, hold political office, etc. -- or they aren't. There is no useful middle ground in discrimination, so when race is used as a basis for it, it has to be a discrete classification, not a continuum.
Real biologists do not use the term "race" because the concept is bullshit. They use "sub-species", but, as you wrote, it applies to populations that are biologically able to interbreed, but have different gene frequencies because they have been kept separate by geography or other reasons (such as disjoint flowering times). When the obstacle disappears, subspecies usually mix and the distinction disappears.
Indeed, natural evolution invented sex even before it invented legs or brains -- because it found that mixing genes is good for life in general. By nature, individuals generally have a drive to pick prtners for "DNA mixing" outside their group, as long as the genetic differences are small enough to allow it. Avoidance of other "races" is always a social imposition.
No human population has been isolated long enough to make cross-breeding impossible. The Australian aborigines and the Andaman Negritos, who may be the extreme branches, have split out from the trunk less than 100'000 years ago. Even the Neanderthals are now known to have interbred with "modern man" ("Cro-Magnons") in Europe and elsewhere.
For as far back as we can tell, entire nations have migrated by thousands of kilometres in a few generations, because of war, climate changes, population pressure, hunting opportunities, etc.. Mountains, rivers, glaciers, deserts, even open oceans were never hard barriers to human movement. Humans are very mobile, so genetic flow between populations has never been zero.
Moreover, each gene spreads, mutates, and is selected for mostly independently of other genes; so there is no single gene, or even a gene combination, that could be used to distinguish the so-called "races". A single individual who moves from one population to another will inject his genes into the latter's gene pool, and any of his genes can spread to the whole population, just by random genetic drift, over the span of a few centuries. So, even if there are environmental or social factors that make dark skin (say) disadvantageous, that trait may quickly disappear, while other genes that came "in the same boat" with dark ski will persist and spread.
A sobering exercise, that underscores how silly the notion of "race" is, is to compute how many ancestors you had by the year 1000 CE. Any one of your genes may or may not have been inherited from any one of those ancestors. How can you tell that none of them were from "race X"?
Well, I have read a cubic meter of Scientific American and half a cubic meter of Science (which is the second most prestigious journal for biology, after Nature), mostly cover-to-cover; and some books on human genetics, like Cavalli-Sforza's. I have dabbed in computational biology and even published a some minor things on that subject. Yes, I think I can tell with sufficient authority what scientists think about the concept of "race".
That is not a strong enough reason to explain why scientists reject the concept of race. There have been many "scientists" and even real scientists in the past who have openly held racist views, and even got praise and money for that. Around the 1930, Stanford was sort of a center for that sort of thing. (I recall that in the 1980s, William Shockeley -- one of the inventors of the transistor -- was at Stanford, and was an outspoken believer in races.)
But, curiously, the belief in "race" seems to have been stronger among scientists from other fields than among geneticists. Even before the genetic code was deciphered, geneticists could not ignore the complexity of actual gene distributions, that had little correlation with racial boundaries. Now that we can read the genome, the absurdity of the concept is obvious even to non-geneticists.
Ancient astronomers assumed that the stars were located on a sphere centered on the Earth; so they though that the apparent groupings of stars on the sky were all important, especially those across the ecliptic that were "visited" by the Sun, Moon, and planets. Astrologers built a complicated intellectual edifice on top of them. But once astronomers determined the true distances to the stars, and figured out the three-dimensional "geography" of the cosmos, they realized that constellations and star magnitudes were just meaningless illusions, and astrology was total bullshit.
Well, the concept of "race" among biologists had a somewhat similar history...