Were Black soldiers from the WW2 and Korean War era able to fully?
In theory, yes, However it pays to remember that Jim Crow laws had not been obliterated yet in many parts of the South. I have a hard time imagining a Black WWII veteran attending the University of Alabama at that time. Clearly the remedy was to relocate to states with a more congenial racial climate. It also pays to remember that the WWII generation, as well as the majority of the Korean War generation, were children of the Great Depression, with many being from Dust Bowl states. This meant that many from the poorer sections of the country did not even finish High School. Of course, the GI Bill also included training in trades schools, but some would have needed remedial training before attending one of those. So, fully utilizing the GI Bill would not have been easy for minorities or Rural poor whites.