An escaped pet wouldn't last a month in the wild. Hunting is a skill taught by mother, to kitten. A pet would likely also have clipped nails.
Sure that pet cougar killed a 70 pound, tied up dog last summer. And Cat was fat, probably not very fast, if that dog were free to run, may have been different.
Sure a wild dog can get by in the wild, but they usually live near dumps and places that they can scavenge. And many of them are born in the wild. They not pets that have been tied to a 30 ft Chain there whole lives.