One of the more important things to consider, IMO, when discussing "race" in the context of D&D: are there things that the "races" of D&D (or any other game) can do that nobody outside that group can?
Consider in Mass Effect. Asari are monogendered, they're all "biotic" (psionic, in D&D terms) to at least a limited degree, and they can have kids with almost anybody (unless they have a genetic condition which causes them to kill their partners during mind-sexins.) They also typically live to be at least 10x as old as humans today; millennia-old asari are the norm. (It's also not completely clear, but implied that they're mammalian like humans, whereas the other council races definitely aren't.) Then you have salarians, amphibious, extremely short-lived, but with an incredible metabolic rate (which makes them move, think, and speak faster than most other races) and fantastically little need for sleep, just a single hour per (Earth? Standard?) day. An old salarian is in his 40s (and most of them are male because they're haplo-diploid), but in that time he has had, using Earth timekeeping, over 336,000 active hours; compare this to a typical human that sleeps 8 hours per day (slightly under 263,000 hours), and couple it with almost certainly doing more than humans do in the same amount of time, and it's not hard to see why the salarians are not especially worse off for living less than half a human's lifetime (even in ME, where humans typically live to be over 100). Krogan appear to be effectively immortal unless slain, with numerous organ redundancies and, apparently, sufficient genetic adaptability that the Genophage begins to be attacked by gene-repair mechanisms less than 2000 years after it was applied.
But let's circle back to D&D, and particularly the edition I prefer (and thus am the most experienced with): 4th edition. Something actually rather old-school* about 4e is that there are, in fact, particular things which only certain "races" can do. Elves, Ealdrin, and Drow are all considered to be branches of the same phylogenetic tree, yet there are discrete, innate abilities which each can use that the other two cannot. Eladrin can all teleport, innately; it's not weird or special, it's perfectly normal for them. Elves have preternaturally acute senses and accuracy, such that when they really want an attack to land, it's very likely to land. Drow can create temporary supernatural darkness (to which they are immune) or inflict a sort of dark Faerie Fire effect, hampering their enemies and preventing them from hiding. All Dragonborn either can breathe an element or instill terror into their enemies, and become even more driven when the chips are down. Humans are impressively versatile, capable of being "among the best" in any measure of ability, and can also master both more of the fundamentals of their disciplines while maintaining interests in many fields (extra skill). Half-elves, meanwhile, are either able to merge elements of two disciplines into a harmonious blend, or able to find an edge for themselves or their allies in almost any bind. Dwarves aren't just extra-hardy, they're better than any other race at drawing on their internal reserves and bringing the pain back to their enemies, and heavy burdens don't weigh on them as much as they do on others. Devas have ancestral memory. Tieflings can instinctively counterattack--with infernal flames.
I could go on, but that paragraph was getting long. The point is: every 4e race has a clearly discernible, inborn ability, and in general those abilities are intended to be purely a function of biology (Dragonborn, maybe Elves), psychology (Humans, maybe Devas), or....for lack of a better term, "thaumatology," the innate magical nature of a being (Drow, Eladrin, Tieflings). It doesn't matter what culture a Dragonborn is raised in, they'll either have a stare that could kill, or some magically nasty halitosis. In that sense, all of the above-listed races really do have a practical, observable effect which makes them different. Some, like dragonborn, are really biologically distinct, such that it would be very difficult to sell them as the same, but others like Elves/Eladrin/Drow are in theory about as different from each other as human ethnic groups.
So...I can understand why, given the fraught nature of the word "race" when used about real people, someone would be uncomfortable using it in a game. But at the same time, even those who don't like it recognize that (a) the "races" in many editions of D&D really do have some traits which aren't shared, and (b) it's hard to come up with a term that isn't inaccurate, excessively broad/narrow, or just as problematic.
That said, there is one part of it that would be nice, if we could extricate it: separating what is clearly biology (e.g. Dragonborn breath weapon) from what is clearly culture (Dragonborn +2 History and Intimidate). If we can do that--if we can have Human physiological traits be stat flexibility and, I dunno, save bonuses or something, and then have various cultures which are typically but not exclusively Human--I think it would go a long way toward easing the tension some people feel about the word "race." It would emphasize that race is not culture, that it is possible for "an elf" to identify more with the mostly-human-and-tiefling nation they were born in than the mostly-elf-and-eladrin nation their parents emigrated from. The fact of their biology as "an elf" would remain, just as the fact of a dragonborn's biology would remain regardless of where she was hatched, but that biology need not entail other parts of their identity.
*It's sort of a hearkening back to the race-as-class deal, but with more modern sensibilities.