In fact, most racist Democrats DID either switch to the Republican Party because of Civil Rights, and Richard Nixon, who in the 1950s had had a very strong civil rights record, not only encouraged it but made it central to the Republican Party. This is so well known that only someone disconnected to reality could deny it. The Republican Party would nominate three opponents of the Civil Rights Act for President, and two of them won: Barry Goldwater, Ronald Reagan, and George H. W. Bush.
What is NOT as well known is that there was ALWAYS a strong racist contingent within the Republican Party starting shortly after the Civil War. Chester Arthur signed a law banning Chinese immigration to the US. William McKinley called for the “Christianization” of the Philippines; he was so ignorant of history that he didn’t realize that there had been multiple cathedrals in the Philippines before Jamestown was settled. Theodore Roosevelt’s attitude towards American Indians was that they were almost subhuman. William Howard Taft and Herbert Hoover tried to expel black people from the Republican Party in southern states — a major reason why black people started voting Democratic. Calvin Coolidge openly sought KKK support in 1924 and signed the racist immigration law that would doom millions of Jews to death in the Shoah; two Republican KKK members were elected Governors that year, in Indiana and Colorado.
Things did get better in 1936, as the Landon/Knox campaign made the first national outreach for black people since the 1870s, and in 1940 the Republicans nominated former Democrat Wendell Willkie who was a true believing passionate supporter of civil rights. Dewey, Eisenhower, and Nixon (in 1960) were also supporters of civil rights, but then the party went in the wrong direction while the Democrats as a national party were expelling their racists.