And that survey (like many in the early days of DNA analysis) only looked at mitochondria or the Y chromosome, that are inherited from father or mother respectively and thus are easier to analyze.

First, there is no such thing as "white genes" and "black genes".  There are genes that produce the black skin pigment (an essential protection against sunlight), which are broken or inhibited in people with "white" skin.  There are other genes for eye color, hair color, hair wave, etc.  The frequency of those genes is very different among populations that have lived for millenia in relative isolation in northern Europe and Southern Africa (which can be explained by well-known environmental factors, as well as by random drift).  Hence, indeed, when individuals from those populations are brought together to the same place, the differences are visually striking.  But There are many more genes whose effects cannot be seen and do not significantly affect survival.  There are also many variants of the same gene whose effects are hard or impossible to see visually, and both populations have mixtures of the same variants.    

Some of your so-called "white" genes are indeed recessive (like blue eye color), but others are dominant, or have partial effect when two variants are present.  But a recessive variant of a gene is not "lost" when it gets paired with a dominant one: the children of two back-eyed persons may be blue-eyed.  Ditto for black or blond hair, curly or straight hair, etc..

The important point is that linkage between genes is limited and weak.  Therefore, when a "white" population mixes with a "black" one, the first generation children will have one copy of each kind for each gene, but after a couple of generations there will be all combinations -- WW, WB, and BB -- for each gene independently.   But the independence of gene mixing, by itself, would imply that racial classifications are meaningless, so "racial scientists" have conveniently developed a blind spot at that place in their minds.

As for that picture, it proves only that Americans have a high incidence of some peculiar genetic defect in their visual system, that renders them unable to perceive gradations of skin color and other "racial" traits.  A functioning non-American eye will see that the child at left has a mixture of skin color genes that results in a light brown skin, surely much lighter than the skin color of her African slave ancestor.   And you can be sure that she has close to a 50% mix of genes from her two parents, just like the child at right (apart from mitochondial genes and those in the X chromosomes, which they inherited from the mother alone).