I guess it's all social once we inject language into things but to me it seems like either an attempt to obscure or a debilitating need for precision.
You've hit the nail on the head. When we use terms like "physical basis" or "scientific," precision is absolutely required. One can generalize ethnicity or race based on physical features as a useful tool, just as one can explain the relationship between a nucleus and electrons using the orbital model, but that's a representational model, not a scientifically accurate one.
An excess of melanin may tell you generally that a person may be non-European, but it's not a reliable measure that compensates adequately for biodiversity, e.g. is an albino non-African? Is a Bengali African?
The reason race is socially constructed is because culture and climate play a heavy role in the development of behavior and even phenotypical changes. What value we assign to those differences is solely social in nature.