One would presumably watch all the other animals chowing down on critters lower in the food chain, and think "hmmm, maybe there is something in that!"
All the early hominid hunter gatherers managed to figure it out ok (and chances are we would not be here to even have this discussion had they not!)
Does fire not work on your desert island?
Or put it out of its misery (depending on the severity of injury).
At which point, I can't see how even a vegan could with clear conscience object to eating it. The death was premature, but at least it can still help support life afterwards rather than go to waste.
I don't require any cognitive dissonance, since I don't consider rearing an animal for food production to be cruel or to necessarily cause it suffering[1].
We are giving it a more comfortable (if shorter) existence than it would likely have "in the wild" with its food and medial needs attended to, and then a swift and hopefully painless death at the end.
I appreciate that you don't see it that way, but we will have to agree to differ.
[1] I am not claiming that animal cruelty never happens - although from what I have observed, well regulated professional farmers are highly unlikely as a group to mistreat their animals.I think you will find many males of various species will take any opportunity to get jiggy with the girls given a chance!
(in fact man is the only species that can choose not to reproduce)
I have relatives who are (well were - retired now) farmers. They always took great pride in the care of their dairy herds. Oh, and my Grandfather was a master butcher with a chain of shops in the east end :-)