I believe this problem with the word "race" which biologists have needs to be handled as a communication problem. The way that biologists use the term is, like the word "species" Darwinian, but that is not what "normal" people mean. The flexible concept of an interbreeding population is fine and clear to me, but it is not what most people think of when they read about biologists proving the existence of races and species.
I think this is what prompted most anthropologists to jettison the word. But then there are two strains in anthropology that are hard to reconcile with each other. One strain rejects the word “race” with its unpleasant social correlates, but pretty much retains the nineteenth-century concept. Another strain rejects the concept of race entirely.
This of course becomes confusing because we can see statements like “anthropologists all agree there’s no such thing as race,” but in fact some really do believe there are no such groups, while others believe in such groups for all intents and purposes but refer to them only with Orwellian terms!
I don’t have any answer, really, but you’re certainly correct that it’s a public communication nightmare. In my classes, I estimate half the students just assume that anthropologists are lying about race.