I agree, there are no easy solutions. But changing culture to 1. become more aware of implicit biases based on background (aka: "become more woke") and 2. actively integrate marginalized peoples into common socio-economic institutions do work as difficult, effective solutions to reduce inequalities based on these categories. So any critique of them would have more weight if there were viable alternatives.  

I am not convinced that the right-wing resentment comes from treating people differently, but rather from the fact that what were club goods and services increasingly have become public ones available to everyone. It's not very different from say 1832 U.K Conservatives being angry that The Reform Act of 1832 made political institutions more equally inclusive by including more people of relatively marginalized classes. And yes, the effect is that many of those Conservatives loss political-economic resources they (or their ancestors) previously guarded for centuries, in the same way historically dominant classes (white, men, etc) lose previously guarded socio-economic resources when their club goods and services become more public - but that is an effect of reforming or restructuring institutions so that people who were previously excluded are more equally included. 

Treating people as individuals who are independent of their backgrounds or the groups they're embedded in would make sense in a society where everyone was born with the same opportunities and all results came from their own chosen decisions. Unfortunately, that doesn't describe our real-world societies. In our real-world the social institution of inheritance allows historical injustices to propagate beyond their historical period. In order to correct the effects of those historical injustices you either eliminate inheritance(s) and give people equal opportunities upon birth or you aim to holistically consider the opportunities individuals inherited due to certain groups they're embedded in when considering the merits of their choices continuously throughout your interactions in life. 

And this makes practical sense too, to consider background. A poor person who has had fewer opportunities like tutoring and motivate parenting who achieves the same test scores as a wealthier person with more resources of this kind, likely had to make very different (merit-worthy) choices to achieve the same result. Likewise, when it comes to other marginalized classes. So it's not merely a matter of "treating people better or worse" due to their status in a social category, but rather what they have achieved despite a lack of inherited opportunities. 

If inequalities like different races having more or less rights are left in the past, then people will equalize through the fact that white people aren't actually smarter or better than say black or brown people, nor are men actually smarter or better than women. Those who inherited more from times when some had less rights will squander it if they're incapable.

Life will never be perfectly "fair" because human beings are all different in our strengths and weaknesses as individuals and in the choices we make. The best we can do, in my opinion, is give people equal treatment and let them sink or swim on their own.

Why do you think they'll automatically equalize without intervention? Here in the U.S -- discrimination on the basis of race has been banned for more than a half-century and many metrics of social-economic status have not fully equalized between people of different races (some have exacerbated), and that is with active positive-discrimination and a cultural shift for society to become more aware of implicit biases. Without these things, progress would be even further behind. In fact, the regions of the U.S where things have equalized the most (ironically many states in the U.S South; excluding the deep south) have done so because they were forced to through active legislation like the Voting Rights Act and forced school integration. The states/cities that didn't have Jim Crow laws actually are less racially equal (in many metrics) today than many Jim Crow states (again excluding the Deep South) because they weren't singled out in the federal legislation and weren't actively monitored and forced to integrate.

For example, income inequality. 

Or school integration 

Or segregation in general 

You really do need to actively change culture and the structure of institutions for equalization to occur. It doesn't happen automatically. The equal abilities of individuals don't matter much when talking about supra-individual entities like social institutions. 

As for the last paragraph, life never being perfectly fair is not an argument against making things fairer. Otherwise no social change (including the aforementioned political ones) could have ever been motivated. 

The institutions already offer equality though, it's been illegal to say exclude black people from a University or pay women less than men for decades.

Outcomes are created not just by external forces but by internal ones as well.

The bolded is false. It is made with the assumption that just because people are nominally equal before the law, the social institutions and forces treat them equally in fact. But that obviously isn't the case. 

Otherwise, how do you explain the gaps in outcomes? What are the "internal forces?" If you believe there is no difference in the group-level abilities between races, and social institutions de-facto treat people equally on the basis of race, then at the racial level outcomes should be the same or rapidly narrowing. There is no reason they shouldn't according to your premises, because there is no explanation for the differences. But we obviously don't see that. We need an explanation for the differences in outcome, and the evidence points toward there still being a difference in the treatment of people, as well as the resources and opportunities available to people on the basis of race. 

With pay and gender for example, a woman or a man get paid the same amount if they work they same job for the same amount of hours, yet men typically choose higher paying jobs that are for instance dangerous or unglamorous. That's an internal force, driven by choice.

There are per-occupation wage gaps that can't be explained by this. Also I am not convinced this is an internal force that isn't impacted by socialization according to gender. There is also evidence that as fields shift towards being female-dominated from being male-dominated that compensation decreases, and the converse. 

But to stay on the original topic, what is the internal force for racial-outcome differences? Why is the U.S South more racially equal and integrated than the U.S North and Midwest despite it being more historically ideologically racist? How do we explain this without considering the historical impact of active social and political interventions? It can't be something like culture because the variance in African-American culture isn't along regional lines. 

Socialization and culture are a big factor, yes.

I don't live in the US nor do I have any experience with the African American community specifically so I cannot make any comment there, but one thing I do see in my own life as someone who came from a poor rural town is that often in demographics that fare worse in terms of socio-economic status is there is a culture of learned helplessness, where people accept their status as unfixable rather than striving to escape it, and are socialized into behaviours that perpetuate their own poverty just cos people naturally imitate the behaviour of others around them.

For instance, the kids of teenage mothers tend to go on to have kids in their own teens in turn, or people piss away their money on things like alcohol or gambling cos it's what their friends do, or people drop out of school cos the culture around them doesn't value education, all of which are choices which contribute to poor outcomes.

Again, this doesn't explain the regional differences. We are looking at metro-areas (non-rural) within different regions of the U.S.

In two regions (North & Midwest) the Federal Government didn't actively enforce school integration, didn't actively employ Black Americans as a policy initiative, didn't construct public housing policies with integration in mind, and didn't create Voting Rights Act districts to enfranchise Black voters. The Fair Housing Act (FHA) was also very loosely enforced in this region, which enabled racial segregation through mortgage loan approvals (or disapprovals) aka redlining. 

In the other region (South), the opposite of this happened. There was active enforcement of school integration (through military force and bussing), there were government contracts compliance programs that disproportionately affected the South due to large shares of federal employment of Black Americans in the region, the Voting Rights Act districting mandates meant that politicians had to be more responsive to Black voters (who made up a much larger minority share of the population), public housing policy aimed for increased integration, etc. 

The cultural differences across racial-lines exist in both regions. If anything Black-Americans in the South were in a poorer position starting out than those who fled to Northern cities during the Great Migration. So they should have had more of the "learned helplessness" you are suggesting might account for the difference. And the South has the additional issue in that historically it was a much more ideologically racist region. 

The answer really is that strictly enforced structural desegregation policy made the racial gaps in the South smaller than in the North. 

https://ideas.repec.org/p/nbr/nberwo/3894.html

https://www.aeaweb.org/articles?id=10.1257%2F0002828042002679

https://s4.ad.brown.edu/Projects/Diversity/data/report/report2.pdf

It's not 1954 or 1965 anymore though, most people alive today weren't even born back then, segregation and such have been over for decades and decades.

Giving people a free ride cos of the skin colour or their gender doesn't solve the problem, at some point, people have to help themselves.