They had the post WW II economic boom to take advantage of and that blacks were largely prevented from taking advantage of.
Not all of them.
Irrelevant. All that matters for the discussion at hand is that blacks had the obstacles that whites had, plus many more and much greater obstacles impeding their ability to benefit for the post WW II. This is an undeniably significant causal factor responsible for the lower incomes, less job related skills, and less education of blacks on average than whites on average.
That actually counts as a pretty strong data point too: it's been my experience that black families that at present time find themselves solidly in the middle or lower-middle class are usually descended from those that managed to get in on the post-war boom.
So then you are acknowledging that how well people do is heavily determined by the situation that they are born into, including their parents income, education, and general social resources available to them. Good, that is a start, because one cannot acknowledge that fact and reasonably deny the extreme disadvantage that most blacks were born into compared to most whites. Although the size of the disadvantage and the % of blacks to which the term "most" applies have both steadily decreased, that disadvantage remains on average and the magnitude gets bigger and bigger to a very extreme degree as you go back for centuries.
The most common method is through the GI Bill: black soldiers returning from the war were able to get college educations and subsidies that opened opportunities they never would have had otherwise. The second most common seems to be public sector jobs (e.g. teachers, postal workers, public transportation). Union jobs was another contributor, but not all unions actually were inclusive of black workers (some of them still aren't).
I am not claiming that blacks had zero opportunities, but very few of them compared to whites and almost no higher paying opportunities (e.g., blacks who earned a college degree via the GI bill were still not even considered for most jobs requiring a college degree). The kinds of opportunities to get any real wealth out of the boom were virtually non-existent for blacks. The rise in their average wealth from 1940 to 1960 was much smaller than for whites, and was due to fewer opportunities and just down right legal prohibitions on what they could do.
The primary determinant for success are the number and variety of opportunities to succeed. Whites coming out of depression poverty had many times more opportunities to improve their situation than blacks did
By the 1970s, the momentum from the postwar boom had started to dry up; the oil crisis siphoned a lot of strength out of the economy, and and black workers who were already marginal -- both by tradition and by lack of specialization -- were the first to feel the burn. Those families that were already middle class by then were able to survive until things started to improve in the 80s; those that were still blue collar got pushed WAY to the back of the line and ended up in the ghetto.
I agree with all of that. What that means is that black had about 10 years of anything resembling legal equality and freedom, before things went to shit. And since most of them, no matter how hard working and skilled, would have still been at the bottom rungs of "Blue collar" they got especially fucked.
Did many whites also get fucked? Yes. Are many whites today in their 20s to 40s children of economic struggle with a mediocre education, uneducated parents, and no inherited money or other resources? Yes. Are people who claim that all whites are "privileged" and should feel so wrong? Yes.
But it is critical not to ignore that the effects of slavery and racism are still a/the major cause of why black communities on average have less education, less wealth, less skills, and more crime. This is true even though there is variability in those communities and some people do more with what they have than others, just as is true of every group. Also, because physical and social communities are still centered around racial lines, even a black kid born with middle class parents is still likely to live in a neighborhood with more crime, worse schools, more friends getting in trouble, and fewer friends with any chance to go to college.
This is a mechanism by which a disadvantageous environment can be basically "inherited" independent of one's personal family wealth and education level.
Again, there are are % of whites born into much worse situations than many blacks. Policies and ideologies that ignore this are destructive, unjust, and racist. However, rational beliefs about why group level differences exist must be cognizant of the many direct and indirect factors that have and do create more obstacles for blacks on average.
How all this related to the OP is that, we must recognize that racism exists (something that the OP has yet to show anyone denies), and that the effects of racism continue to this day and will continue for generations, impacting group-level differences and creating more obstacles for the average black person than the average white person. However, it is fallacy to go from this to concluding than any bad outcome for a black person or any and every disparity between a sample of whites and blacks on any given variable is evidence of racism, especially racism by any of the more immediate actors in the situation rather than residual effects of historical racism. Making this inference to specific situations requires more direct evidence and all rational consideration of evidence requires consideration of alternative explanations for that evidence and asking "given everything else we know, would we expect to observe the same data even if no direct racism was responsible for the outcome?" This modest request for rational evidence-based thinking is what the OP objects to and is the real perspective that the "Anything but Racism" argument is a dishonest misrepresentation of.