Better hunting techniques can significantly raise Malthusian limits.

First, you have to remember that old-fashioned humans were one predator among many: improved hunting techniques could raise our share of the pot, as well as decreasing other predators' tendency to eat us. Also, modern humans seem to have used carcasses more efficiently than Neanderthals: they had permafrost storage pits and drying racks, so could have preserved meat for long periods. Neanderthals didn't, and I think they must have wasted a lot. Next, moderns used snares, traps, nets, bows etc to catch smaller game not much harvested by Neanderthals: they also made more use of fish and molluscs. And lastly, more plant foods. Altogether, their innovations gave them a larger share of the game, used that share more efficiently, tapped marine resources (lots of salmon in Europe), and harvested resources at a lower trophic level ( plants for example), which are always more abundant.

Hunting to extinction happened in some places, but not everywhere: it hardly happened in Africa at all. It happened most in places with no previous hominid occupation.

Implied population densities are, I think, extrapolations from known hunter gatherers, salted with some archeological info. Estimated densities range from 0.01/km sq to 1/km sq, strongly dependent upon resources. A lot like those of bears. Lower than chimps, probably: but then chimps manage with a lower-quality diet than humans. We're probably not as good at digesting leaves as they are.

Probably you have to consider Pleistocene climate as well: the world was generally nastier - colder, drier, lower plant productivity due to low CO2 levels.