I’d argue the Black community’s targeted and strategic preference for national Democrats caused the shift in attitudes on civil rights for both parties, rather than any inherent shift in the parties’ rhetoric causing a huge change in Black voting habits. In other words, the fact that Black voters largely abandoned the GOP in favor of the New Deal very clearly influenced how Republicans chose to view civil rights in general.
I 100% agree that causation gets reversed here a lot and your interpretation is closer to right. Northern black voters moving towards the Democrats in the 1930s and 1940s made Northern Democrats dependent on them and concerned about their votes, and Northern black voters flirting with returning to the GOP in the 1950s made Northern Democrats really, really intent on trying to hold onto them by strengthening their commitments to civil rights. The fear that the black voters that helped make Democrats competitive in NY and PA and IL and MI might start going back to the GOP really put the fear of God in Northern Dems on Civil Rights.