Wild predators don't consume more than the environment can sustain

Sometimes they do

Wild animals aren't less taxing on the environment per se, they're just differently taxing. Long living ecosystems are likely to be closer to balance. But never perfectly balanced.

Domestic cats are just cats same as the wild ones, but in different numbers and locations

For example, people like to talk about beavers as good for the environment. Sometimes they are, sometimes they aren't!

https://www.youtube.com/watch?v=r1wwtGgnzg4

Leaving cats to find their own game instead of feeding them farmed meat is significantly worse for the local population (of birds in particular)