Maybe three people from Thurmond's party ever switched. They retired not soon after. As for the Southern Strategy, it failed. It served it's purpose during Nixon's campaign, in 1964 and 1970; but it didn't hold much weight in 1976 as the entire south voted Democratic in electing Jimmy Carter. In 1980 they all voted for Reagan, Democrat and Republican alike, 1984 was the same way, and so was 1988. In 1992, Bush and Clinton carried the South equally, as well as again with Clinton and Dole carrying 6 states apiece. And by 2000 and 2004 Republicans had regained control of most state legislatures in the south for the first time in history. Suffice it to say, the Southern Strategy had a negligible impact on politics post Nixon.
And actually, I was responding to the question. As you may already know, the Klan was revived in Atlanta, in 1915. To say they didn't have political influence in the south is preposterous. There were people like Arkansas Congressman James M. Hinds, who was assassinated by the Klan in October of 1868. Then the Klan subsequently they murdered close to 1,300 Republican voters that year, all of this which resulted in a Democratic landslide in Columbia County, Georgia during the gubernatorial election. While it may have only been a social group for a year or so, it sudden became a paramilitary organization with political aims and goals, helping Democrats, hurting Republicans. They knew which side to join, and they stuck with them in the South.
The Political Influence of the Ku Klux Klan in Georgia, 1915-1925 by Clement Charlton Moseley - JSTOR
You're still pedaling this false equivalence bicycle. There's no doubt the KKK persecuted Republicans, but they didn't do so because they were Republicans or because they represented some Conservative ideology; far from it. They did so because those Republicans (at the time) represented change to the social order that the KKK sought to preserve. That is in fact how the Klan saw itself.
Before the Southern Strategy, being a Democrat was simply the established entrenched custom. The first Republican President had vanquished the South in a Civil War and it was bitter about it; there wasn't a whole lot of chance that President's party was going to gain a foothold. So for the foreseeable future, the Democratic Party (the only one that had really existed before the War and still did) was "it". If you were a racist asshole KKK member who wanted to run for Senate, you ran as a Democrat. If you were a kindly old soul who wanted to run for Dogcatcher, again you ran as a Democrat. Neither of those makes the DP the party of either racist asshole Klan members or kindly old animal lovers.
Although the rural areas have morphed like a Christmas color wheel after the CRA, in some places the pattern continues to hold even in recent time. One particular self-absorbed asshole wanted to be mayor of a large city -- even though he was a lifelong Republican, he ran as a Democrat to get the job. His name is Ray Nagin. But again, his changing his party overnight doesn't make him have a different political philosophy. It makes him either an opportunist or a pragmatist, depending on how sympathetically you want to view him. But it didn't make him an ideological Democrat.
Political parties are tools, the means to an end, the end being power. They like to assume certain political leanings as a mission, that's true. But that doesn't make them some kind of religion that pervades everything any member does or says. It isn't.
And if the Klan were to re-rise today (it's happened before) they would be persecuting Democrats -- for exactly the same reasons they formerly persecuted Republicans. Which just demonstrates how not about a party it is.