“....people of South America were thriving before and after the middle Holocene...” So did they hibernate in between or something? :-)
Probably most of them died, there was some small population that managed to make it through, then their descendants started to thrive after the temps rose. Same kind of thing happened in Europe during the Little Ice Age -- famine killed off a bunch, the Black Plague followed intermittently, wiping out whole villages here and there, but claiming the very young and very old, disproportionately. One difference (besides degree, probably) is that the survivors enjoyed literacy.