First, look at the way it's done in the wild. Prey animals are viciously attacked by gangs of ravening wolves or lions, or by solitary hunters like tigers and jaguars. This is not quite as bad as it sounds, I don't want to drive a horror spike into anyone's hearts. It's safer for the predators to do it this way and the targeted animal suffers less pain when it is attacked in a way that induces shock, which causes numbness and euphoria, then quick death.
Predation is absolutely necessary. Every animal can outgrow its food supply. Caribou and deer strip vegetation down to nothing, then they die slowly and painfully in ruined and diseased ecology. Some contagious disease may even be a reaction that protects the ecology. Carnivorous species had to develop to protect the vegetation so that herbivore populations could be more stable.
It's the same thing when humans kill a certain number of herbivores and eat them. Fur and leather get more use out of the carcasses, and grinding up mink to use as fish feed, pet food, and fertilizer is using them for food. Industry doesn't like to waste usable biomass. It's more profitable then gold. Most likely if an environmentalist can think of a good way to use biomass, it's not only been thought of by industry, industry already has a better idea because that's the way that industry lives. Boiling down bones for gelatin and glue is a long-established art and that's getting close to salvaging everything but the squeal.
When humans slaughter animals for food, we have to be pretty bad at it to be worse than nature at the slaughter, and we don't leave wounded animals to die of infections and starvation weeks later. I personally don't even want them to die, but it's a necessity that has to be faced. Better to face it in an orderly fashion and with protection for the animals and the humans involved.
Nature kills off old and sick animals to make room for young fresh ones. Reproduction has to be maintained at some reasonable pace to keep up genetic variety. We can't go around forcing people to stop breeding animals on account of we're too successful in the job of keeping them alive. Nature usually kills off the majority of animals before they reach their first year. Humans are miraculously effective in keeping them alive, which fits our nature. Humanely killing selected animals for population control is a very acceptable alternative to allowing the lot of them to be subjected to slower painful deaths from diseases, parasites, and crippling physical and genetic illnesses.
Killing for population control should be an acceptable method and it should be actively disallowed as an issue in these fights to pass breeding bans or controls. Thus far we still have the right to decide which animals are pets, which are breeders, and which are food or culls, and this is a right that every animal owner must insist on. It is very nearly the only button that AR factions can consistently push. Without that they have less than half the power to legally disrupt animal owners. They use their own killings of animals in shelters as leverage against everyone else. They also use the deaths of animals when the legislation that they pushed for forces those animals out of loving homes. They use this issue in bad faith to damage every animal-using industry. We must take this issue out of their hands.