I disagree. If that were the case many of those smaller species would have gone extinct ages ago due to natural predation from wild foxes and ferrets. Domesticated cats are a new issue because they often hunt with the same voracity of wild animals, but don't experience the natural pressure on their population that wild predators do because they're human pets.

If there were as many human households with domestic foxes as there are with cats, then foxes would absolutely be considered an unnaturally potent problem for wild rodent and bird species too.