I've only had cats and dogs, not goats and birds like you, but I understand what you mean about cats influencing the others because they do it to my dogs, too.
Do you still have a lot of cats? <<<
After I got the wild cat out of the house, I have not had another house cat, except babies that I nursed and found homes for.
But I feed about 150 pounds of cat food each month to the wild cats, all kinds and colors, they come to dinner.
I don’t have any other pets now, I did once, when I could take care of them.