Of course it exists. There is venison and other wild hunted meat. In many places these animals have to be shot anyhow for population control. Overhunting would be the only issue here.
Then there are purely pasture fed animals. Especially with goats this is common . But there are also some cattle and pig farms. As long as the land itself wasn't deforested and is given enough time to repair itself, it's perfectly sustainable.
Then there are things like keeping chicken in your garden that are only fed kitchen scraps. Depending on your household size you can even keep 1-2 pigs that way.
If any of these options are available to you, they can be more environmental friendly than some plant based foods. Locally sourced version is definitely better than having plants shipped across the globe.
As you said the only issue is the quantity and also the desire for premium cuts. A lot of meat is currently wasted because it's "undesirable". Some parts will find their way into animal food but a lot also just gets thrown away.