The margins show more D's voted for the bill than Rs.
And that's with the Dixiecrats voting against it.
Democrat JFK called for it in 1963. Democrat LBJ  made use of his experience in legislative politics, along with the bully pulpit he wielded as president, in support of the bill. Your assertion that it "was a politically strategic move that LBJ et al supported once it was clear that blacks were going to be integrated and given the vote, anyway (thanks to Republicans,") is a farcical lie. The Democrats controlled both the House and Senate by big margins. The Republicans couldn't pass anything on their own.
As for D switching to R, I'm talking about the voting patterns in the South.
Look at this map. It's overwhelmingly Red. It used to be Blue.
https://en.wikipedia.org/wiki/United_States_Senate#/media/File:115th_United_States_Congress_Senators.svg
Nixon's Southern Strategy worked.