It takes 10 years to get a logging permit
Only in the USA.

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Agreed - this would definitely be a good step but it seems highly unlikely given the immense number of introduced species (insects, plants, mammals, fish, crustaceans, birds, reptiles, amphibians), all of whom have or have the potential to out compete/kill native species.
But back to the OPs original question - I completely agree that it is the cat owner's responsibility to control their animal and if they don't, the chicken owner has to protect their livestock/pets - by whatever means necessary. I just think that it is reasonable to try speaking with the owner first and then escalating from there if it produces no results. There is no reason to sit by and let your chickens be killed if you have proof of who the predator is. For what it is worth, I have also had non-lethal luck with a paint ball gun used on a neighbor's cat who was causing issues for my flock.