What are the roots of white privilege and why is still considered a thing?
I am sure many of you will simply downvote or reply "tl;dr". That's ok.

I'm just offering little primer about the institutionalized foundation of racial inequality and its long lasting legacy whose effects still linger today. This is not all inclusive as it excludes the effects of Jim Crow laws in the South following reconstruction.

“40 Acres and a mule” was a real thing and occurred (briefly). Andrew Johnson, Lincoln’s successor and a sympathizer with the South, overturned the Order in the fall of 1865, and, returned the land along the South Carolina, Georgia and Florida coasts to the original owners”

The 1866 Homestead Act included black Americans and encouraged them to participate, but rampant discrimination, systemic barriers and bureaucratic inertia slowed black gains. Plus, even the low prices and fees were often too much for the applicants to afford.

New Deal
The National Recovery Administration,not only offered whites the first crack at jobs, but authorized separate and lower pay scales for Blacks

The Federal Housing Administration created in 1934, was to insure loans for construction and repairs of homes. White middle-class families could buy suburban homes with little or no down payments and extended 30-year amortization schedules and their monthly charges were often less than rents the families had previously paid to housing authorities or private landlords. But the FHA had an explicit policy of not insuring suburban mortgages for African-Americans

The 1933 Agricultural Adjustment Act (AAA) drove many black farmers from the land. As subsidies were paid to (usually white) landlords for not growing certain crops on a part of their land, black (and white) sharecroppers and other tenants were the first victims of the new policy. The evicted farmers were often forced to migrate to northern cities as the southern countryside had no alternative to offer.

The GI Bill helped white Americans prosper and accumulate wealth in the postwar years, it didn’t deliver on that promise for black veterans. In fact, the wide disparity in the bill’s implementation ended up helping drive growing gaps in wealth, education and civil rights between white and black Americans.
While the GI Bill’s language did not specifically exclude African-American veterans from its benefits, it was structured in a way that ultimately shut doors for the 1.2 black veterans who had bravely served their country during World War II, in segregated ranks.

All this time blacks were receiving inferior education through the principle of “separate but equal” (see Plessy v Ferguson). Black schools were certainly not equal in terms of facilities, books or faculty.

Now It doesn’t take genius to see that, while whites were allowed to build equity and advance their education, blacks were simply struggling to achieve their “Inalienable rights” to be treated equally.

I might add that many programs which could have been beneficial for blacks were fought by powerful Southern Senators and Representatives, Democrats who would later form the foundation of the Southern GOP.