When I was a kid, the Democratic party included "Souther Democrats," who were all the things you say they were. History records that Strom Thurmond, Richard Russell and other Southerners left the party for a while. In 1947, President Truman ordered the armed forces to be de-segregated. In 1948, he included a Civil Rights Plank to his Presidential Election Platform. The Southern Democrats had none of that, and broke away, Forming the Dixiecrat Party with Strom Thurmond for President and Fielding Wright as VP they ran independent of the Democratic Party, for many years. While they did call themselves "Democrats," they opposed all civil rights legislation.
The so-called "Liberal Republicans," which included Nelson Rockerfeller and Everett Dirksen delivered necessary votes to get civil rights laws passed. When they delivered the Civil Rights Act and Voting Rights Act in the mid 1960s Lyndon Johnson predicted the Southern Democrats would forever sever their ties with the Democratic Party. They did, only to be lured into the Republican Party by Richard Nixon's "Southern Strategy. Strom Thurmond left the Democratic Party in 1964 and helped recruit Southern Democrats to join him. Strom Thurmond, the racist leader of the "Dixiecrats-States Rights Party," was a Republican Senator for 39-years, when he died in 2003.
So, if the people who made the Democratic Pary the "party of slavery, as you correctly note, became Republicans, what did that then make the current Republican Party?