I don't see it as simple,
1 What is the number of different races out there and how the race is measured?
2 Where is the line between mixed and non-mixed race? (percentage?)
3 At what point new races stopped to develop and how long we are going to keep lumping new born people into the existing groups?
(sounds almost like some sort of social construct...)
White (not a scientific term) Americans are the descendants and a mixture of different European ethnicities(?), so are the white Americans a mixed race/ethnicity or should they be considered as their own ethnicity? When the term "white" started to be used, it excluded some light skinned people like the Irish and still today, Russians and ethnic Jews. Harrison Fords parents are Jewish/Irish descent, so is it wrong to call him "white" then? What if he "identifies" as white himself?
You like Asians, as in Pakistanis etc.? Brits calls them Asians, Wikipedia calls them Caucasians and Americans calls them Arabs, but what does the race science say?
I see race as a continuum, with various groups denoting some prevalent traits, but without open gaps between them. So you can't say that someone is, say, 100% Asian. But someone who is, say, 90% Asian and 10% Caucasian would overall be likely called Asian.
When I say that I like Asians, I really mean Eastern Asia: people originating in countries such as China, Japan, Korea, Malaysia, Thailand... The exact classification really varies between different scientific and cultural groups.
I believe your characterisation of the argument is incorrect. The argument is really as such: "There is greater genetic variation within populations than there are between different populations, therefore biological race doesn't exist."
Note that we aren't deliberately singling out individuals that are more different to prove a point, as in your characterisation. Instead we are looking at the broader statistical variation that occurs within and between populations.
Several studies have found that, of the total genetic variation that occurs between humans, around 85% is found within populations. https://en.wikipedia.org/wiki/Human_...etic_variation
I'm not sure, however, whether this alone means it's safe to conclude that biological race does not exist, but based on numerous other arguments including the ones made in the OP's video, the idea of biological race is at least very dubious.
Consider two Gaussian distributions, one centered at 1 and having the STD of 10, and another centered at 1.1 and having the STD of 10.1. There will be probably around 100 times as much variance within a large sample from one distribution, than there will be between the two samples, but the samples still comes from different distributions.
This is how I see race as well. Within one races the variance can be very large, while two groups from two races put next to each other will differ much less - but that difference will still be traceable.