Wouldn’t matter if it was a doe fawn, or a B&C buck, there is always a rifle at hand in my house and coyotes would be getting murdered.
Wouldn’t matter if there wasn’t a deer, there would be dead dogs in the field.
You miss my point though (and by you, I mean tokguy specifically), a deer walking past you in the timber, as a sub trophy animal for discussion sake, with a broken or missing lower leg, is going to end up exactly the same way regardless of your interference or not. The only thing that will change, is the when and how.
No matter what, that deer is still going to die, probably within a couple km radius, and coyotes are going to get some or all of it. Ravens are going to get some or all.
Nothing about that deers life will change, except the length, should you choose to end it. That deer might live 1 more minute, or 4 more years, and there is no way to know. It might have a healthy, productive life but for the fact that you choose to interrupt that.
If you choose to let it walk past, it may or may not survive for some
length of time, and you really don’t know for certain how long that will be.
That said, I have ended the suffering of dozens upon dozens of animals that had very little chance of survival and it merely sped up
the inevitable. But I know that the only thing that it did, was lessen my feelings of empathy for the plight of the animal. Nothing changed for the animal, other than the length of time it took for death to arrive.