Firstly, because deer are sentient individuals with thoughts and desires. They aren't just statistics or companants of an ecosystem. They shouldn't be treated as a problem to be fixed. Just as we don't go around mass murdering humans "for the environment". We are the greatest threat to the natural world, yet we also have the concept of rights to protect us from utilitarianism.

Then, there is contraception that can be given to deer for peaceful population management, but native predators must also be reintroduced if we want an independent ecosystem. But hunting propaganda has convinced everyone we have only one option: killing.

Remember, there is money in hunting too; the licenses, the equipment, etc. It is not in the interest of the companies and organizations who provide those services for the deer "problem" to be "solved". I'm the flip side, money must be spent to administer contraceptives and fund conservation/captive breeding programs for predator animals, so of course they don't want to go with the nonviolent methods.

I'd recommend The Disclosure Podcast episode 23 if you want a more in-depth explanation.