You've got to remember that, in those pre-expansion days, St Louis was not only MLB's westernmost town but also its southernmost and that radio station KMOX, 'The Voice of St Louis', broadcast Cardinals games to not just Missouri but Arkansas, Texas and beyond. So there was a certain degree of the Cardinals feeling like they represented the south even if not all of them were from it. The Redskins were the same in the NFL and they stayed white only until into the '60s* (not that you'd ever hear the NFL acknowledge that) because their games were carried by radio into Virginia, the Carolinas and beyond well before there were teams in Atlanta, Florida, Charlotte, etc.
The Cardinals finally integrated in the early '50s after Anheuser-Busch bought the team. The story was that when 'Gussie' Busch toured the team's training camp he asked where the black players were. When told they didn't have any black players he asked, 'Why not, we sell beer to black people why wouldn't we sell baseball to them as well'. So maybe not an altogether humanitarian gesture on Busch's part but a practical one to him which in turn led to the same result.
* I once saw some pictures (probably in a magazine though I can't remember) about the Redskins one of which was of a teenager/young man carrying a sign demanding to KEEP THE REDSKINS WHITE. I wonder if that guy ever realized the irony of his wording.