The chronology of that association doesn’t work. Democrats were staunch opponents of civil rights until the 1960s. Even then, more republicans supported the Civil Rights Act of 1964 than Democrats.
Even in 1976, Carter won with the traditional new deal coalition of southerners and northeast progressives. He won South Carolina by the same margin as Massachusetts. Even in 1980 against Reagan, Carter performed much closer to Reagan in the south than his overall national performance (outright winning Georgia, but coming close in most southern states).
The current alignment of southern states with Republicans really dates to the late 1980s to early 1990s, when Republicans started flipping Senate seats in the south.
Pinning this on Democrats’ support for the Civil Rights Act two decades before is one theory, but another is the industrialization of the southern economy. Southern New Deal Democrats represented agricultural states against protectionist northern industrial interests. But look at the economy of places like Georgia and Tennessee today. They’re built on low taxes and low regulation to draw companies away from high tax high regulation northern states. That started happening in the 1980s. Georgia is a really archetypal example of this: it’s probably the most openly pro-business big city in the country, and has a famously cooperative relationship between Atlanta Democratic mayors and Georgia Republican governors centered around attracting companies from places like New York: https://www.ajc.com/news/state--regional-govt--politics/with...