Except that the "big change" was not modern humans. Neanderthals and modern humans (and denisovan humans too!) had been interacting for a pretty long time before the last glacial period. The neanderthal population of the near east is probably why sapiens' first established Eurasian populations came from the Indian Coast, rather than through the near east - sapiens migrating the northeastern route were either rebuffed or absorbed by resident neanderthal populations.
(interestingly this means that neanderthals are the reason sapiens was not only the only human to reach Australia, but also why we did so as early as we did. We just hugged the Indian Ocean coastline all the way down)
The "Big Change" was that the last glacial period wrecked the Neanderthals. They had survived (but not exactly thrived) prior glacial periods, but the last one was especially harsh. it reduced the European neanderthal population to isolated "islands" on the fringe of glacial deserts, and scattered the near eastern and North African populations due to the multigenerational glacial drought that would kick-start what we see when we look at the middle east today.
What this means is that when human populations - neanderthal, denisovan, and sapiens - all began recovering and re-expanding durign the modern interglacial, sapiens simply had the largest and most genetically diverse population of the three, due to living in warmer ice-free territory during the glacial period. That means we were the one that did all the population-absorbing, unlike during hte prior interglacial, where our small migrant populations were getting absorbed.