Why would an increase in racial mixing necessarily lead to LOWER inequality instead of being an unrelated factor?
That question seems to lean on an assumption that is part of the racist system--one of those things where people who seriously do not think they are being racist at all are having their thinking channelled by a racist system. In the OTL US racist system we assume that everything we do that is not explicitly and obviously a matter of interracial relations is therefore something "white" people would do just the same if no African Americans ever existed at all. This might or might not be true but we have to think very very carefully about it, because the fact is USA society as we know it OTL is deeply shaped by the whole slavery/Jim Crow/ modern apartheid interaction.
So it is not automatically reasonable to assume that successful liberation and preemption of Jim Crow leads automatically to the "default white" American self image of OTL, the one in movies and so on where you just don't see African Americans at all and everything is written as though they never existed.
That hardly proves we would all be better off--speaking from a common person point of view here--and certainly this is not "necessarily" true. But is it anyway a reasonable outcome?
Well, consider the infamous comparison of American working class people compared to European. In Europe, for the past half century or so, the overall standard of living of working class people has been clearly quite good compared to that of USAian people. Time was the USA's overall advantage in hegemonic shares of global wealth, our leading position in technology and so forth, our clear global leadership position both economically and politically, all fudged the fundamental disparity of political power of American working people versus European with the collective US advantage offsetting the basically weaker position of Americans as working people--in fact arguably in the WWII-to say 1980 period where Americans took it for granted we had the best way of life in the world, the political power of common people was then greater than it is today too.
But since the 1980s "Reagan Revolution," it has become quite plain that the US working person is in a stagnant rut economically speaking, not sharing in the surge of overall wealth which has accrued entirely to smaller elites, and in this context the difference between European social democracy and American laissez-faire ideology has come down on us hard.
Why this contrast? Why should US working people be the tail end Charlie losers in the developed world on both economic and political fronts?
In this context I think the effect of the USA being a highly racist society is highlighted. It was and is the racial divide and rule of our domestic society that sets us up for this.
Therefore it seems reasonable to me that if an ATL USA stumbles into a situation where the self-interest of African Americans can check the ability of white supremacists to repress them and set them up as a subcaste, that rather than simply erasing some embarrassing chapters of historic narrative from our consensus history but leaving broad dynamics unchanged, this would in fact change the basic dynamics so much that the USA must subsequently develop rather differently.
The difference might be subtle perhaps, but there is every reason to assume changes would all be in the direction of for the better.
Another line of argument I take as evidence along these lines is the contrast within the USA between the general positions of working people in the North versus the South. You seem to be taking it for granted the former overwrite the latter and that is all, but again, if we are honest and take a hard look at how US society today works across the board, and has in various phases throughout our history, the whole Union is in fact strongly affected by the differential treatment of African Americans across the board. (Abolitionists of the more radical type realized this early on--for a popular but I think fairly shrewd presentation see Harriet Beecher Stowe's
Uncle Tom's Cabin--throughout the book she makes points emphasizing the basic unity of northern and southern "white" society and gainsaying the comfortable notion that consequences of the "peculiar institution" were confined peculiarly to Southern people--and despite her narrative stance and possible top-of-her-head conviction that African emigrationism "back to Africa" was the favored solution, her book also deeply underscores the integration of African Americans into American society as we know it. So the contrast in outcomes for poorer "white" people between North and South is really in the context of both being in turn part of a larger divide and rule system, but the contrast does remain, and was and is stark.
So it seems plain to me that American racism (in this manifestation--I'd say there are at least three distinct though interrelated American racisms, repressing African Americans, Native Americans, and immigrants respectively, each with quite distinct dynamics and methods, though all integrated in a deeper framework serving harmonized purposes, and we observe crossovers in the actions and words of people both perpetuating and opposing them. At least three--probably we should break out a few more, such as anti-Semitism and bigotry against Latinos as such, though one could try to subsume these into the three basic categories I offered) serves as an effective means to place checks on the basic radically democratic ideology of US patriotism grounded in the rhetoric and thus implications of the US Revolutionary War, the Pandora's Box of the Declaration of Independence and the spirit of the Gettysburg Address. Insofar as US bigotry tending to repress African Americans is short-circuited then, it seems reasonable to me that by and large, American working people will be better off in general through a number of concrete channels.
These boil down I suppose into two trends--on the one hand, stronger, more vigorous developments of the potentials of the bourgeois revolution, in the form of less initially stratified economic development on capitalist terms. The South is much less, perhaps entirely freed of, such legacies as sharecropping; not only African American but also poor "white" communities have the potential to indulge in a mix of mild ad hoc practical social democracy and vigorous private enterprise. Wealth will tend to spread out broader and deeper and be even greater than OTL, at least insofar as grassroots rather than centralized capitalism is responsible for US economic gigantism.
And as the suck tide of centralization comes into play, the political aspect of centralized bourgeois power in the form of state support of the privileges of great private wealth versus the masses has a less covered field to run on. Small communities with autonomous republican institutions of a scale intended to be accessible to direct democracy along the lines of a New England town meeting (not saying this will be the universal form of communities, but I have assumed the ideology advocated for this scale at any rate, and recommended the operation of at least partial mass meeting direct democracy, which various communities might take up or not) the community level is one where individual tycoons cannot evade the scrutiny of their actual neighbors. On the Federal scale, with no intermediate layer of several dozen large republican states, centralizing policies overriding the grassroots populism in the small communities must be justified and defended in the plain light of day, with the eyes of those dismayed by any centralist rulings turned directly on the Capitol with a mind toward whom the citizens will vote for in the next election.
It is then somewhat up in the air just how persuasive the sorts of Spencerian Social Darwinist reasoning that pervaded the Gilded Age OTL would be here. For a long time, quite a conservative notion of laissez faire under Jeffersonian-Jacksonian assumptions might prevail, but lacking the two level fan dance of bouncing questions back and forth between Federal and state wheelhouses, central rulings on what is and is not permissible would be made on one stage, in one forum, with the eyes and ears of interested parties on both sides turned in that direction to watch and judge. This in turn puts pressure on the judicial dimension of these resolutions of issues.
Fundamentally it boils down to, will the Marxist sketch of republican government in a liberal-capitalist age as in fact the steering committee of the ruling classes, and hand down whatever rulings favor their interests, or will mass republican democracy work as advertised as a clearinghouse of all interests with enough numbers behind them to bring their claims to the floor of the houses of Congress?
To suggest as I am that short-circuiting one of several operational forms of US racist bigotry will improve matters does not mean that there will be no such bigotry in operation at all, though I certainly am saying there will be less. I think there is good reason to believe the situation will be better, if not perfected. It is also possible, looking at it from a more Marxist side, that capitalist hegemony will find a way. Perhaps, sensing or clearly reasoning that removal of the state level of the US republican system would rob them of a vital shelter of their interests, such interests being powerful and essential to a ruling Communalist-Republican Civil War/Reconstruction regime, will insist on yet more clauses added to the new Constitution that enshrines interpretations that OTL were embedded in court decisions to, as Oliver Wendell Holmes protested in a dissent, "enact Marshall's
Economics." And this in turn might give shelter to white supremacists and make the overall picture darker than I thought it would be.
The OP, to get back to that, asks what happens if the OTL Republicans decide to perpetuate Reconstruction nanny state dictatorship over the South. I am guilty of pretty much dismissing that premise as impossible, on the assumption such differential treatment of the South as a region cannot be sustained long, certainly not on the basis of a purportedly democratic republican system. I am critical of the assumption implied that the South is fundamentally and eternally different than the rest of the Union--to be sure if we gamed out how northern and western political dynamics could work out to sustain such a long dictatorship, lasting to say 1920's per the OP, the fact of the differential treatment will tend indeed to accent regional identities. I turned it around and applied a radical and permanent Reconstruction to the whole Union, common policy for all.
If we stick to the premise and assume both that the Union excluding the South has the political will to sustain special Reconstruction indefinitely ("merely" going to the 1920s is after all a half century right there) and that either the regional differences are baked in or that the very act of holding the regime in place perpetuates them, such that any eventual withdrawal of this authority would surely lead to Jim Crow reaction at a later date, these are assumptions I think should be questioned. Or actually the OP just specifies the former and might be fishing for views on whether the latter holds or not. I do think extending Reconstruction longer might do some good, but clearly offset by the harm that separate and invidious treatment does. Probably to get a different result than OTL, different forms of Reconstruction or different conditions, rather than merely drawing out what was done OTL, are needed. Some here seem to believe that just by doing perfectly feasible different things, the outcome might have been quite different and I broadly agree with that, but am not sure just what it would take. I do think that the ability of African Americans to defend themselves in their own communities is probably essential. A less radical suggestion might hit on how to do that within the framework of the traditional states, without this perhaps crazy Communal-Republican radicalism I never put on a firm historic basis.
But in any TL where at least one of the many functioning American bigotries takes a fall, even remains but diminished, I surely do think everyone is going to be better off to some measurable degree. This includes even the beneficiaries of divide and rule racism, for the kind of benefits they get are short run, a matter of "playing it safe" in the immediate situation. I think if such factors are shorted out, the long term commonweal will be improved to a degree that even those who lose in the short run by having the rug pulled out from differential accumulation of wealth and power will be offset by a higher overall standard that leaves them individually no worse off, or if drastically less wealthy and powerful than OTL, at any rate still above the higher average. They might then be absolutely worse off despite avoiding collective costs that leave the OTL nation worse off but them better off, but I would still argue for the intangible but real benefits of being modestly rich and powerful in a stable secure society versus being oligarchic aristocrats in an unstable one prone to either general strokes or revolution.