The problem with the racial classifications we commonly use is that they have no scientific basis.
Instead of 'race is only a human construct', a better characterisation would be 'race is a pseudoscientific construct' or 'racial classifications are not based on sound science'. In which case Nye and Dawkins would probably be in agreement.
Your characterization implies that there is zero covariance between who is placed in the same or different racial sub-categories under the common approach and who would be in different or same sub-categories under a scientifically valid biological approach. IOW, the current categorization doesn't match a biological one any better than if all people were just randomly assigned a sub-category.
In more concrete terms, it assumes that two black men from Alabama are no more likely to be genetically similar to each other than either is to a white man from Alabama.
Do you think that is a defensible assumption?
If we removed information that coded for skin color, and gave geneticists the DNA of all Americans categorized as "white" or "black" under the common rubric, would they be able to do better than chance at categorizing the DNA into the 2 subgroups?