I’m surprised you listed feral cats!

Feral cats are the offspring of domesticated cats that have been let out of peoples homes’, perhaps they’ve escaped or more likely were thrown out and because they were unspayed and unneutered they reproduced. I’m sorry but I’m a proponent of TNR, Trap Neuter and Return of these cats to their colonies. They can sometimes be tamed, as I’ve tamed and socialized several into loving housecats. Usually it takes much patience and because of not enough homes they are best off in their feral colonies with a caretaker providing food and care.
As far as killing non native species, the domesticated cat is not native to the US, it was brought over from Europe during the early settlements. Should we therefore kill all cats?
The horse also is not native to the Americas., it was brought over from Europe in the 15th century. Its domestication had a profound impact on the native Americans. Much like the feral cats, wild horses still run in the west. Various organizations have adoption programs available to try to adopt out these beautiful animals rather than turn them into the glue factories.
Your quote as far as killing by any means whether by poison, gun or trap has a potential for error. If you do HAVE to kill, make sure it is done HUMANELY. The animal did nothing wrong to you. There is no reason to inflict needless pain and suffering.