All of the "irresponsible" ownership of dogs, cats, or other animals does not cause as much problems as the rampant stealing by humane societies and owners of "scamtuaries" under color of law.
Our worst dangerous dogs and our worst dangerous animals of other species kill fewer people each year than almost any other cause of death and thousands of times fewer than mistakes made by doctors. Even without improvements in the sense of responsibility of the average human, we do not have a significant problem there.
A really strong bout of the Black Plague would overwhelm our medical system in modernized countries such as the U.S.A. The common housecat saved us from the plagues that decimated Europe. They save billions of tons of food from rodents each year. Small predators are necessary to prevent rats and mice from destroying homes, crops, stored foods, and bird populations. This means that we risk human and animal lives when we "clean out" feral cat colonies.