@Ken-1122 said "tend to gravitate," which you certainly know is not an assertion of an absolute condition, so your rhetoric implying otherwise is disingenuous.
But different cities have evolved differently, because there is always more than one thing happening.
Urban areas, particularly in the North, were often segregated by old ethnic enclaves from the early 1800s--Irish, German, Slavic, Italian, et cetera--and then with the Great Northern Migration in the early 1900s, black.
Racist redlining that was applied to the black enclaves was not applied to the white ethnic enclaves. That meant property values in those white enclaves were not artificially depressed...people could get home improvement loans, have new homes built, and other things that improve property values and encourage the more wealthy to remain. Moreover, redlining also prevented black areas from being purchased by black GIs under the GI home loan bill or by wealthier blacks, so there was not the post-war middle-class infusion that other areas enjoyed.
Eventually, though, white ethnics lost a lot of their ethnic uniqueness and later generations that were wealthy enough to move to the suburbs did so. Those areas often deteriorated into impoverished white urban areas. Walthier blacks also moved out to the suburbs.
Smaller northern cities that had neither white ethnic enclaves nor the great influx of blacks in the Great Northern Migration often did wind up with integrated poor areas of both blacks and whites.
Southern cities and towns, where segregated housing was more strictly defined, tended to evolve somewhat differently. From the very beginning, the "white part of town" had both its wealthy and its poor areas, both being distinctly separated from the "black part of town."
The black part of town also had both wealthier and poorer people as well (because of segregation), but still property values for wealthier blacks remained depressed for all because of racist city polies that restricted city services and schools in those areas. After the Civil Rights era, wealthier blacks began moving out to the 'burbs and into formerly all-white areas that had better services and schools.
It's probably worth some study to discover why lower-class whites seemed to have discovered mobile homes while lower-class blacks seem not to have done that...but that is an observable phenomenon.
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