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I think you actually conceded the OP’s point when you moved away from average to sub-set. :-)
"But many people say slavery and segregation, though wrong and evil, are not to blame for racial disparities. I don't understand the logic behind this."
In the 1970s there were many Vietnamese boat people that arrived in the United States.  They were quite literally foreigners, did not know the language, had NOOOO money, and about 18 years later a number of valedictorians at Southern California High Schools had last names like Ng or Nyguen.  In one generation, there were children of Vietnamese boat people that became doctors and lawyers.  From prestigious schools.
Clearly if America has deep racism this would be impossible.  How did they do it?  Well, their families stayed intact with both parents.  They lived in very small apartments, or multiple families would live in one house with individual families sharing one bedroom.  And those families vauled education and pushed their children to do well in school knowing that education was the path to prosperity.
" it's clear that slavery and segregation had long lasting effects that have lead to the disparities we see today."
Then why are the people of Appalachia generationally poor?  They are White, why is their income on average about the same as Black Americans?  I will ask that question again, why are Appalachians that largely come from broken homes with absent fathers, high drug use, and without a big cultural push to educate their children out of poverty have wealth about the same as Black Americans?
In the United States, if you A)graduate from high school (and can do high school math, and read at a high school level), and B)get married before you make babies, and C)stay married, then you will not be poor.  Approximately 97% of the people that do those steps are not poor in America.
Lack of wealth especially the lack of being able to build wealth from one generation to another has more to do with poverty, and the things that cause poverty, than then things related to race.  The Vietnamese boat people proved that it is possible to THRIVE in the United States, by adopting cultural attitudes opposite of poverty.
So, when you say "But many people say slavery and segregation, though wrong and evil, are not to blame for racial disparities. I don't understand the logic behind this."  There is your answer. >

I think one question to ask is "did the coping mechanisms available to a group facing hardships having lasting utility?" With that question in mind, it becomes clear that the coping mechanisms available to Black people (eg, don't excel or you'll get sold and separated from your family, you may have to act violently to protect yourself, etc) were harmful to them in the long term. A lot of norms/behaviors formed during slavery still exist in some form today (sugar/fat in most cooking; appreciation of physical ability, etc), and they are now counter-productive. 
Compare this to Jews, whose coping mechanisms after the Holocaust allowed for education and going into business (they were often limited to jewelry and banking), but where the general coping mechanisms from that time are extremely useful into today's world. 
Same with immigrants/Asians. 
Your Appalachia example fits this as well. What coping mechanisms were available to them? Not good education. Not saving money (with the "company stores). Not improving their communities (extractive capitalism). They had booze, drugs, and sex available to them. The only coping mechanism that worked was to leave all together... >

If the Appalachia example fits here then that undercuts that poverty is based upon rascism from the OP. >

I think there can be many reasons for poverty. Economic and educational factors contributing to poverty in Appalachia doesn’t negate those same factors plus racism contributing to poverty among Blacks. >

When the Title is:

The disparity between the average white family income and the average black family income are the result of slavery, segregation, and racist lawmaking and the long term effects of slavery, segregation, and racist lawmaking

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And your answer is "there can be many reasons for poverty."
And additionally

Economic and educational factors contributing to poverty in Appalachia doesn’t negate those same factors plus racism

Then you are conceding my point.
the OP was not "The disparity between the average white family income and the average black family income are the result of a bunch of economic factors that also plague Appalachia and also slavery, segregation, and racist lawmaking and the long term effects of slavery, segregation, and racist lawmaking"
It was "The disparity between the average white family income and the average black family income are the result of slavery, segregation, and racist lawmaking and the long term effects of slavery, segregation, and racist lawmaking"
The fact that there is a sizable cohort of white Americans that have a similar average family income as black families is evidence that it was not slavery, nor segregation, nor racist lawmaking as the reason.