We live near a creek, albeit a small one, and something struck me as I was watching various species of ducks and geese coming and going. The small, hobby farms, with fences, that are around mean that any migratory mammals are blocked from being in the area. Therefore, the most we see are birds - in small flocks.
Watching them come and go, it struck me that the small numbers in each flock mean that, were I to start hunting them, I would probably only get a couple of birds, and the migratory group are more likely to avoid the area - particularly when I'm around a lot.
If there were large groups, and I wasn't living right next to the creek, hunting might be worth doing - there would likely be more chance of a good catch. The short answer is not fouling one's own backyard. If I wanted to successfully hunt, I'd better go hiking to somewhere I do not live...
It's a question of short-term versus long-term. Short-term, hunting would be okay, but not so much in the long term. If I want the long term, I'd be better taking the time to do the slow gathering that is finding eggs, not getting much in the short-term, but more dependable, and less hassle.
There is a national park near here, and apparently there are wild deer populations around. I don't know enough about deer husbandry, but it might be worth at least learning something.