I don't know all of these, but I suspect the list may be over broad in who it considers Conservative. Al Gore Sr ( father of Al Gore the vice-president) was very much a New Deal Democrat, a Progressive and not a Dixiecrat. While his record on civil rights was varied, ( he did not support the 1965 Civil Rights act, but he also did not join the Thurmond and sign the Southern Manifesto , or call for "segregation forever" like Maddox and Wallace) he very much opposed the Vietnam War ( while insisting that his son, Al Jr, join the army fighting it). And he indeed lost his Senate seat to a true Republican conservative, Bill Brock, who painted him as too liberal for Tennessee when the South swung in Nixon's favor in the 70's.

The website's statement ,that it was Republicans who supported Civil Rights for blacks and Democrats who opposed it, is occasionally made. It does have a bit of truth: the South was very much dominated by Democrats after the Civil War ( the Republicans being the party of Lincoln and the overseers of Reconstruction). And Theodore Roosevelt was remarkably progressive on the issue, as opposed to Woodrow Wilson who was not. But th statement is overly broad: it is also true that the big Supreme Court decision of 1954, Brown vs Topeka Board of Education was not used by by the Eisenhower administration to begin desegregation efforts in the country, and Eisenhower as President did little for civil rights.