I don't think the composition of races in humanity is really all that different from the time of the ancient Romans thousands of years ago, except a small new segment of mixed people. Why would it change in only 500 years?
There is some evidence that skin tones respond quite quickly to selective pressures. Geographic isolation is a key part of skin tone stratification, after all, and skin changes may not require more than a few millennia to change
within a population. Given the mixing rates, the movement toward the middle would be -- will be -- comparatively swift. That said...
I think the most accurate answers have already been given, though. We're not far enough into the future for that kind of homogenization. ME1-3 would only be about 8 generations from today, or about 4 generations from the elder generation still being around to see the Reaper war. Expecting fringe cases to be eliminated quickly is far too short. If we take U.S. population (67.3% non-hispanic white), 4 generations of completely random mixing will still leave 4.2% of the population with purely pasty bloodlines, or a third at least 3/4 white (incl. hispanic). I use U.S. because of its lack of a dominant native population, thanks in part to colonial genocide. We still have quite a lot of population isolation right now, including large swaths of isolated land such as central South America and politically-isolated populations such as North Korea. All of these divisions will push forward the time scale required for more complete mixing. Furthermore, there will inevitably be more isolating factors. The people who reach Andromeda will ultimately become a distinct population (and possibly new species) from those who remained in the Milky Way.
Of course, if they were going for realism, it's fairly obvious that the next protagonist should probably be born into a world that is comprised of a lot more mixes and less like turn-of-the-millennium Edmonton. After all, in the United States, half of all Asian women marry outside of their own ethnic group. However, that's not to say that BioWare are really doing a terrible job or anything, because it's really hard to find a mix for the audiences of 2016.
This would never happen because evolution doesn't work that way especially since we have recessive traits. Evolution favors diversity (so long as they can remain in the current environment). If there is no diversity among our DNA there is no evolution.
Evolution causes diversity, not the other way around. There is no primordial diversity pool from which all diversity arises; random mutations and sexual reproduction will establish diversity and adapt a species to its ever-changing environment. Recessive traits are not necessary for adaptation, nor are recessive traits phenotypically unexpressed (heck, you're probably more thinking epistasis anyhow, which can also inhibit the expression of dominant genes). Some traits can be both adaptive and quite deadly. Sickle cell disease is considered autosomal recessive even though its associated alleles are codominant: if you have one, you have immunity to malaria; if you have both, you have sickle cell anemia. Additionally, you certainly cannot neglect epigenetic effects. Most phenotypic females have two X chromosomes (excepting Turner with 1, or XY female Swyer or CAIS, or XXX, XXXX, etc), but one is inactivated. Some species more or less express one of the two for their entire lifetimes, and in other species, some environmental effects can influence expression of one or the other. Once again, neither are recessive.
Anyhow, diversity is not limited to raw inheritance of existing factors, and certainly not inheritance of factors currently expressed. But perhaps moreover, lack of wide variance in skin tone absolutely does not preclude genetic diversity. The populations of China are not genetically handicapped because they don't have a lot of pasty white people and dark black people. Ethnic factors that arise as a matter of isolation can be stripped away without hammering genetic fitness in a modern environment. The main questions are likely to be what future selective pressures are, and how those will shape the future of ethnicity.