The question is not whether the working class would gain if they could unite and overthrow capitalism? I hope that is not in serious dispute on this list (with a few quite interesting exceptions).
The question is whether the skin privileged really come out ahead in immediate self-interest by maintaining their skin privilege under capitalism, vs having less skin privilege under capitalism. The answer is not obvious either way, because there are two factors. Lessening of skin privilege under capitalism does mean that (for example) a white male seeking one of the paying jobs usually denied to the non-skin privileged now has more competition. But it also means that he no longer has to worry about a non-skin privileged person's competition lowering his wages. Generalizing, lessening racism divides the pie more evenly, but by increasing the bargaining power of the workers also gives them a bigger pie to divide (at the expense of capital).
It is quite true that the amount of wage flexibility under capitalism is not infinite. But it is not zero either. That is why the empirical question is important. Does lessening of racism under capitalism increase workers bargaining power *enough* so that even those with skin privilege come out ahead.
Note that when true I will never make this my sole argument. To do so would be to endorse white racism when whites do gain from it. But if the weight of the evidence shows that it is true -- it is well worth pointing out. To generalize again: to never show where immediate narrow self-interest and moral outrage or class interests coincide is to reduce yourself to a mere public scold. To focus solely on immediate self-interest and ignore moral or class arguments is to reduce yourself to a mere pork-chopper. (There is an old-time term I came across the other day which deserves revival. I wonder if it was used the way I am using it, though.)
Note also that I do not ask the same question about gender. That is because, most men live in household which include working women, it is almost impossible for men not to benefit personally when economic discrimination against women is reduced. ( I am of course assuming that reducing either racism or sexism increases workers immediate bargaining power to some extent and thus increases the share of labor vs. capital -- even in the absence of an immediate revolution or of infinite wage flexibility. I think this is a reasonable assumption.)
None of this assumes that racism or sexism are solely capitalist plots -- merely shows why workers privileged by either oppression might have an immediate material self-interest in giving up that privilege (and why they might not) and asks empirically which way the best evidence points. The facts given so far are that everyone's wages go up when unemployment among minorities goes down but not as much as when white unemployment goes down. The question is: how much?