Your characterization of events is inaccurate. We have hypotheses about what happened to the megafauna in the last epoch, but nothing proven. Humans probably did hunt them, and it very likely could have played a large role in their extinction. But it wasn't the only factor, nor even remotely the largest. For starters, it was the ice age. It's a pretty safe bet that those conditions would have been the most significant factor in their extinctions.
And it's also the most significant factor when thinking about what humans ate at that period of time. They were pre-agrarian, and living in an extremely cold climate where plant life would have been much more scarce in many times and places. What humans eat in the most extreme circumstances is irrelevant to what we still had the largest tendency to consume throughout history: plants.