The article I linked is a direct refutation of one of Ms. Swain's proof points. There may be others, and I'm quite certain that's likely. If you're interested, seek them out. But let's not pretend that one historian's diatribe on the topic is somehow the objective, definitive truth.
It says nothing about which party formed the KKK or instigated the Jim Crow laws. What Wikipedia calls the "second KKK" was apparently founded by a Democrat, William Joseph Simmons, a fact I was able to establish by checking up on him, also in Wikipedia. The article I've linked to doesn't mention the political affiliations of the founders of the "first KKK", which
...was founded in Pulaski, Tennessee, on December 24, 1865, by six former officers of the Confederate army: Frank McCord, Richard Reed, John Lester, John Kennedy, J. Calvin Jones and James Crowe.
I've also been able to confirm pretty wide acceptance that people who were Democrats started the KKK. However, does that mean that the Democrat party started the KKK? The answer seems to depend what one's political leanings are. Democrats against the notion may say that the erstwhile Confederates didn't vote Democrat because they were actually Democrats, but mainly because they didn't like Republicans, who, after all, opposed slavery and their way of life (in that, by the way, there'd be a tacit admission that it was the Republicans more than the Democrats who were the party of abolition).
I ask myself: did the Democratic party accept southern Democrats into their fold, or did they reject them and tell them to go their own way to form their own party? Well, there's the Dixiecrats, I suppose, but did they form because they'd been told to bugger off, or did they seek to remain affiiated with the Democrats if only by the use of that suffix "-crats"?
Fact is, whatever the case might be, although slavery was abolished, it didn't change the lives of ordinary blacks much because they were subject to Jim Crow laws until the passing of the civil rights act in 1964. That's almost a century after slavery was (nominally at least) abolished. During that period, how many administrations were Democrat? I checked here, and there were five (excluding Kennedy and Johnson). That's five opportunities, a total of around 40 years, for Democrat administrations to make the lot of black people better. The fact they didn't is puzzling if Democrats have always been anti-racist. Of course, the same could be said of the Republicans: why didn't they fix the problem either?
As you said, the issue is nuanced and I'm not a historian, but what I will say is that it's not looking good for the claim that Democrats have alwayse been anti-racist. Especially when there's seems little doubt that Democrats instigated the Jim Crow laws:
Jim Crow laws were state and local laws that enforced racial segregation in the Southern United States. These laws were enacted in the late 19th and early 20th centuries by white Democratic-dominated state legislatures to disenfranchise and remove political and economic gains made by Black people during the Reconstruction period. The Jim Crow laws were enforced until 1965.
I suppose because most black people were in the South...
Jim Crow laws mandated racial segregation in all public facilities in the states of the former Confederate States of America and in some others, beginning in the 1870s. Jim Crow laws were upheld in 1896 in the case of Plessy vs. Ferguson, in which the U.S. Supreme Court laid out its "separate but equal" legal doctrine for facilities for African Americans. Moreover, public education had essentially been segregated since its establishment in most of the South after the Civil War in 1861–65.
It keeps coming back to the North-South thing. Unless the generality of Democrats are willing to completely disavow Democrats in the South (which might include some who were genuinely anti-racist), they can't pretend that they haven't had a long involvement in racism, if not being its prime movers.
There comes a point when if a thing walks, swims and quacks like a duck, it would be perverse to call it something else. One can't forever hide behind a smokescreen of "statistically significant" results and wash one's hands of the weight of evidence.