"DNA is the only source of inherited traits" is an outdated thesis, yes.

Genetic variation within a racial group is comparable to genetic variation across groups, making it incoherent as a biological concept even before you get into more-obviously-cultural constructs like the "one-drop rule".

Large genetic variation within groups does not invalidate that there are measurable aggregate differences in gene frequencies between groups. Which is why it is trivially easy to accurately identify people of African, East Asian, and European ancestry by sight alone.