Dead-on question.
The problem is, when the Democratic Party supported integration (leading LBJ to predict -accurately or perhaps optimistically - that the Democrats had lost the South for a generation) it did not tackle the issue of school finance and property taxes.  A leftist party might have taken this opportunity to propose a national or at least state-based educational finance system that did not promote economic apartheid.  All the Dems did was let the Courts order busing, while millions of northern whites fled to the suburbs, while southern whites fled to white academies, first, and then to white suburbs, later, when they were tired of paying private school tuition.

My point is that the Dems did not examine the class issues around school finance in order to find a position/solution that would benefit all working people everywhere.  They were content to stick with moralism - the moralism of anti-racism - precisely at the point where Martin Luther King had moved on to stress economic issues.