An atlatl which would have been one of the weapons they had, will easily take down a deer.
The throwing club, which was probably our first hunting weapon, is still quite effective at taking out small game.
hahahaha the deer run lot faster than humans. And they have sharp hearing. That is how they survive from a lion attack. They can hear foot steps. A lion which is 100 times more adapted to kill deer than humans, only succeeds in 1/10 surprise attacks. The ancient humans were short and skinny based on bone fossils. They most certainly did not have a daily meat diet.