Well, back in the day, we used to say "there is only one race-- the human race."
And it's important to realise how grounded in white privilege that approach was.
No, it's science. The genetic material that defines what people think of as "race" is an infinitesimal fraction of the human genome. And the variations within these sub-groups is far greater than the variations between them. I recommend Stephen Jay Gould on that topic.
That is really,
really not the point, though.
Variation between individuals is greater than variation between groups, this is true. The "infinitesimal fraction" of the genome that determines what we define as race is insubstantial to group people on anything but the most superficial (and generally meaningless) characteristics, this is true. These variations are nowhere near significant enough to make any kind of predictions or value judgements about a group of people, absolutely. As you say, it is biology. But humans don't function in a vacuum wherein the only thing that impacts us is our genes. Humans have complex psychology, and you can't just disregard it. (And before you recommend I read Gould too, I already have.)
Your position is
indeed one of privilege. It totally disregards all environmental impacts on human psychology. If you'll allow me a generalization, human psychology is about 50/50 nature/nurture. You can't just dismiss that 50% environmental impact, and even if you do, it is still factually incorrect, because racism, unfortunately, is a part of our biology. We inherently show involuntary preference to individuals who look like us. This innate tribalism is the root of racism, which is then nurtured in environment. Innate tribalism isn't a good or bad thing, it's just the way we are. I personally think that accepting and learning about this bias in our thinking is integral to ending racism. Just as learning about our other inherent cognitive biases is integral to understanding the world.
Knowing more about ourselves is a good thing. Denying facts is bad, no matter how opposed they are to our ideologies.
I've never heard a POC tout the "One Human Race" line, because, as
CultCross said, it's a very privileged position. I know that it is not your intent, but imagine how dismissive this position feels to a POC. Firstly, when you are not white you can't just dismiss the reality of race because it bangs you in the head every single day -- and yes, it is a reality; we are social creatures, just because race is a social construct doesn't make it any less real. As I said, 50% of who we are is environmental, of which a huge portion is sociological. Second, white people have a long history of using and abusing others based on their race; dismissing the very real differences in the experiences of a white person and a POC in the modern world is an extension of that abuse.
Racism is bad. We are all agreed on that. Prejudging people's character and capabilities based on race is stupid, wrong, and clearly not based in science, we are all agreed on that. However, dismissing the impact of society and societal constructs on human psychology and behavior is equally wrong and not based in science, and can be as damaging as racism.
"One human race" is a lovely idea, but it is not one that is rooted in biological, behavioral, or sociological science.
There are some authors I could recommend on this topic, if you're interested.