Actually no one heard of your LBJ quote till it came out in a book in 1996, in which someone claimed they had heard he said that on Air Force One in 1963. All the supposed witnesses had been long dead. Prior to 1996, the quote simply never existed at all. Never.
After Reconstruction, the Republicans kind of started to minimize their equality pushing movements. A lot. That's why all those early senator appointees were early. Besides, reconstruction was really about busting the South economically and making it forever dependent on the North.
The Civil Rights and Voting Rights Acts were passed by majority Democrats - NORTHERN Democrats. In fact, if you look you'll see that southern Republicans voted against them. As with everything else you mentioned, the dividing line isn't partisan, it's regional.
LBJ pushed those, btw, as well as desegregation. Heavily.
The KKK was not organized by the Democratic party, that's ridiculous. If that was the case there would not be a Democratic Party today, as it would have been sued out of existence, much as the KKK had been effectively neutered by lawsuits courtesy of the Southern Poverty Law Center and whatnot.
Democrats didn't believe in slavery, they believed it was an issue to be decided by the individual states for themselves. Their views, pro and con, again, were divided along regional lines, causing a split during the 1860 election, and another during the Civil War.
Harry Truman desegregated the military. He was a Democrat.
He also introduced a civil rights plank in the 1948 DNC convention, causing a rift within the party, again, marked by region. The cleaving led to the short-lived DIxiecrat Party. After that went nowhere, Southerners started abandoning the Democrats for the Republicans, who were still very much pro-segregationists.
Hope this helps!