Erstwhile student of human sciences (incl human evolution) here...
First off, the premise of "the stone age" being a monolithic entity is a bit nonsensical (not having a pop at the OP, it's presumably how her kid's school phrased it). It covers millions of years, several species, eleventy billions cultures, and every climate/ecosystem in the world. (Oh yes, and for the animal-raising part of the Neolithic, the whole point was that people mostly spent their animal time raising them, not hunting them.) Which is one reason why the cliched view of "paleo" being eating a load of steak and berries, is bollox. Things would have varied hugely.
Second, it is very likely that hominid meat-eating was a matter at least as much about catching small critters, and scavenging larger carcasses, as hunting big game (another reason why the "me paleo caveman eating steak" thing is bollox--most meat meals would have been things like half-rancid liver).
Third, there are very few groups who can even remotely be called pure, untouched hunter-gatherers. The few that remain are in the most marginal of environments, and have all had extensive contact with outsiders. It's massively dodgy to generalise their ways of life.
Fourth, it's a bit of a myth that HGs prized women having kids and wanted them popping out sprogs as soon as their periods started. Too many people on your land as a HG affects everyone's survival. In the Bushmen, infanticide was fairly common. Girls tended to reach puberty mid-teens, but would not be expected to marry for a few years. In any group, there would always have been women without young children (ie not (yet) married, widowed, they or husband infertile) who would have had no barrier to hunting.
OK, I'll stop the lecture now
I think it's generally recognised now that it's more nuanced than was thought 40-60 years ago. One book that made a big impression on me was "Reindeer Moon", by Elizabeth Marshall Thomas. It's set 20000 years ago--she wrote it having spent time in various close-to-the-earth cultures. The thing that overwhelmingly comes through is that while there are a few principles of life the people have along gendered lines, overwhelmingly their lives were scrappy struggles for survival where everyone just did whatever they could. (Warning: it's pretty grim reading if you are contemplating having children, though!)