A wide variety of animals engaged in persistence hunting (chasing faster prey to exhaustion), including early humans. Of course more efficient methods were preferred if they had the option.
Which would be more than provided for by a full-grown deer... unless the player character has the metabolism of a blue whale. And why does my character go from fully satiated to literally dying after sitting still for a day without food? You can't rationalize this one with "realism", it's actually very unrealistic.
Now the "muh realism" cope is sidelined, we can mobilize the "balance" cope. The game wouldn't be "balanced" if the player could stockpile decent quantities of necessary resources through hunting-gathering. He must be constantly forced to drop what he is doing to gather food until he acquires agriculture because:
Hunting in the real world was far less time consuming than early agriculture. Hunter-gatherers had better access to nutrition than their early agricultural descendants. They didn't switch over to agriculture for ease or convenience, they did so because it provided a higher density of caloric energy in a given area, which allowed a larger population to grow, which could then outnumber and militarily dominate their neighbors.
Interestingly, the Neolithic corresponds with the extinction of the Neanderthal species. Neanderthals were taller and better suited for hunter-gathering, however, they reproduced slower than their Homo sapiens counterparts, who were victorious not because their lifestyle was objectively beneficial to their health and well-being, but because they could outnumber their enemies. And thus they replaced their enemies. I will avoid relating this to modern politics.