Once upon a time I read a SF novel, I think by Larry Niven. This concerned the Frumious Bandersnatch. The Bandersnatch in question was not the invention of Mr. louis Carol, but the result of an alien breeding program to create an absolutely delicious food animal. Only due to some complicated inter galactic politics, the Bandersnatchii were designed intelligent. They were also designed as dirty great worms.
Now these Bandersnatch made a deal with the guys from Earth. Selected humans would be allowed to hunt and Kill the Bandersnatch, in return for a fee. Bandersnatch being big, tough and powerfull, this was not a wholly one sided sport. The Bandersnatch had nothing else to sell for technology to get themselves out of the mess of being stuck on this stupid planet, with no arms for handy things like making tools.
The point is, The Bandersnatchi got what they reckoned was a fair return for a few of them being hunted to death. Maybe humans might make the same choice.
The pheasant is a case in point. Lovely birds. i love to see them in the countryside. But if they weren't bred and released so that they can be hunted and shot, they would not exist wild. Mostly eaten by foxes, probably. So ask a pheasant, and maybe they would agree to being hunted, in return for the chance, however small, of living a free life.
The case of the fox is a little different. Foxes are quite a successful wild animal. They may die nasty lingering deaths from starvation if the population grows too big, or occasional lingering deaths from shotgun wounds. But on the whole they survive quite well. Part of this is that huntsmen deliberately encourage them. To ensure that there are enough to hunt. It might be that a hunting ban could be the end of the fox.