It's not the "South" that got stuck with the Inbreeding meme, it's Appalachia. If you look at most of the "where the inbreds are" memes in each southern state, they point to towards their nearest neighbor that has large parts in Appalachia. As all of you demonstrated and I can add that in Virginia it's West Virginia where the inbreds are.
The reason for this is actually due to a much older ethnic divide, one that predates the founding of the US. Go ask the English where the inbred are and, depending on where in the island it is, they'll likely say Scotland, Wales, or Ireland. You see, the lowland South was primarily settled by the English and in the very old Southern states, often second and third sons of English nobility and the well to do. Meanwhile Appalachia was primarily settled by the Scotts-Irish, people of primarily Protestant Irish and Scottish stock. In fact, the Highlands of Scotland and Appalachia are actually both fragments of a very, VERY ancient mountain chain, thus share a lot of similarities that drew those settlers to those regions as feeling "homely".
Anyway, to say there was bad blood between the English and the Scotts-Irish is... severely underselling things, and much of those attitudes were imported to the New World. The English, being the more cosmopolitan, often would project their worst impulses (things like Incest to preserve bloodlines, etc.) onto the "outsiders" to dehumanize them. Thus, for the English blooded settlers, the Scotts-Irish were said "others" and thus those memes got attached to them.
It didn't help that the dominate economic system of the general South, that of plantation slavery, was completely ineffectual in Appalachia (the land wasn't suited to it, being mountainous and rocky, no big plantations there*). Combine this with the general distaste for things like slavery and serfdom among the Scotts-Irish led to there being a massive economic and attitude divide regarding slavery in the Appalachia vs the General South. Along the Eastern US, many of the Underground Railroad routes that smuggled slaves to freedom moved through Appalachia due to this attitudinal difference.
Basically, Appalachia, while in the South, wasn't considered "southern" by most Southerners and was continually othered by them. Even in the US Civil War, many of the Union units that came from the South came from Appalachian counties of said Southern States, and of course you have the origin of West Virginia there where they were counties of Virginia that refused to secede with the rest of the State. This continued even after the war with how the State governments treated Appalachia as a place to colonize and extract resources from, often enabling the stealing of land from locals by corporations or enabling corporate misconduct in those regions. This was, of course, made more palpable to the general public by dehumanizing people in Appalachia via tropes like them being "inbred hillbillies" who didn't deserve that land and didn't know what to do with it, even though many of those families had been living in those mountains for generations.
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* The one exception to this in formal Appalachia was the Shenandoah Valley of Virginia, which did have good land for plantations and thus saw more slavery than the rest of Appalachia.
