The place where you've taken a wrong turn is "they choose more life." That's anthropomorphization.
The antelope did not lie down and die because it has no concept of life or death, and it was not making a sapient choice for life but rather following instincts which dictate all of their behaviors. This is not the same thing as choice.
Most animals have no concept of their own death ahead of the time it occurs, at the very least not the human concept of death. Instead, they are alive, until they are not. They are aware of the former as an experience (they are experiencing life via seeing things, they are hearing things, they are feeling things, etc), and most of them cannot anticipate the latter before it happens. They are not aware of "life and non-life." They are not aware they are alive, and they are not aware that they will die someday. Most of them aren't even aware of The Self (some are, but most are not). These are human concepts you've assigned to an animal.
They cannot avoid something they have little to no concept of. Instead, they avoid pain. We have documented, scientific proof that animals avoid pain if they can anticipate it, and predators are something which brings pain. Showing pain is likely to bring more pain. Showing weakness brings pain. Animals limp to avoid experiencing more pain. They graze with their intestines hanging out because they are hungry (or because signals from the GI tract are messed up from this injury), and being hungry can cause pain or at least discomfort (hunger pangs), not because they are thinking "I have to eat to stay alive." (and just for the record, "not eating" is like the single most common thing sick or dying animals do. it's like a top tier "this animal does not feel well/is dying" symptom. some random video of an antelope that at this point could have been AI does not change thousands of years of this being a most common and prevalent symptom of animals being unwell/dying. this is how we KNOW that it's not connected to any thought about staying alive, or every sick/dying animal would be horking down food to live longer). Animals give up and lie down to die all the time, but they will attempt to avoid pain in practically every situation.
So we must assume that, if given a choice, animals would choose to avoid pain. Because we see that happening, over and over and over, across all animals we've studied that can feel pain. It's universal. They avoid it, if they can. This is hard wired into every animal with the nerves and functionality to experience pain. The whole reason pain exists is to learn to avoid it, because the things which cause it are usually bad and can cause injury or death. Avoiding pain leads to avoiding death, which can be mistaken as avoiding death, but it's not the same thing.
Which all means that when we know that an animal is in pain, and we know that nothing else can be done to assist them in avoiding pain, the only assistance we can give is a painless death. Doing this IS working in kind with their instincts.
