Genetics is an outdated paradigm and race is not primarily an expression of genetic variation?

"DNA is the only source of inherited traits" is an outdated thesis, yes.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

Race is definitely genetic.

In one particular sense, it is true that there are genetic elements of race and that some genetic diseases are more prevalent among those of one race than another. But once you start drilling down into what we mean by "race", it's not at all a simple picture. For one thing, there are almost no markers that are perfectly diagnostic for race—almost any particular polymorphism is found in almost every race, though at different frequencies. Furthermore, some of what naively you would consider a race (e.g. "Black" or "White") is composed of multiple, genetically distinct populations that are no more closely related to each other than to other populations not in that race.

So most serious geneticists have tended to move away from thinking of "race" as closely linked to genetics—the complicated contours of both population genetics and what we mean when we talk about race just don't fit together all that well, and coupled with a history of unethical policies justified by overly simplistic scientific knowledge in this field, it's best to be careful about the terms we use.

Dogs are to breeds as humans are to race.

The morphological difference between dog breeds are clearly much larger than human races. I suspect the genetic is larger as well, but I have no data to support/deny that claim.

Also, what we think of as race typically has a lot to do with cutural & socio-economic traits. Race isn't a clearly defined concept and might mean slightly different things to different people.

A better term to use would be ancestry. There is a very strong social component to race and ethnicity. Case in point, I know a number of people who "pass" - meaning 99.9% of people incorrectly guess their race/ethnicity and they don't bother to correct anyone. Perhaps you have interacted with some of these types without ever knowing.