I haven’t dug deep enough to offer opinion on that. What I do know is that almost all southern Democrats voted against the Civil Rights Act. All southern Republicans (very small amount though) voted against the CRA.
In the north, most Republicans voted for the CRA and some voted against. In the north, most Democrats voted for the CRA and a few voted against.
Conservatives who spew that talking point know all those details. They’re just hoping to trick people who won’t looking any further into the details into voting their way.