Integrating animals into a food system correctly results in a net increase in carbon retention, even if they themselves produce greenhouse gasses. This is because they help provide soil fertility while controlling the growth of weeds so that trees and other larger, longer lived food plants aren't competing for resources. If the animals didn't do it, you'd need to use pesticides, or a mower, or something else that has more downsides and isn't edible.
There are a lot of permaculture farms integrating animals, fruit/nut trees and crops, but the most famous one is polyface farm (https://www.polyfacefarms.com/).
Ultimately, until agriculture is supplanted by ecosystem creation, and it becomes something that most of the population has some minor role in (if only to save compost for, and help harvest from your local food forest), we're going to be destroying the environment to feed people. Not eating factory farmed meat will slow it down a little bit, but the problem will still be there festering away, until we change our ways or population levels drop an order of magnitude.
There are a lot of permaculture farms integrating animals, fruit/nut trees and crops, but the most famous one is polyface farm (https://www.polyfacefarms.com/).
Ultimately, until agriculture is supplanted by ecosystem creation, and it becomes something that most of the population has some minor role in (if only to save compost for, and help harvest from your local food forest), we're going to be destroying the environment to feed people. Not eating factory farmed meat will slow it down a little bit, but the problem will still be there festering away, until we change our ways or population levels drop an order of magnitude.