Even though I love my cats dearly, I know that they are predators. They will kill smaller animals when they have the opportunity, regardless if they are hungry or not. They are not part of the natural food chain because they are domesticated and their large population is because of human intervene. The cat onwers should be responsible to keep their pet cats indoor to prevent them from hunting in the food chain.
I love cats but I certainly see them as a threat. The feral cat population explodes rather easily, with fewer people being able to afford fixing their pets, or not doing it in time, often causes abandoned pets. If cats are left to roam outside 1 in 3 cats kill at least 2x a week, and with the many domesticated outdoor and feral cats... That can lead to a problem if not addressed.
Domesticated cats kill millions of birds each year. Their hunting activity seriously threatens the biodiversity of bird and small game species in urban and suburban areas. Because so many allow their cats to roam free, they have the opportunity to kill lots of small wild animals. They cannot be trained to ignore their nature unlike dogs.
Cats of all sizes are a part of the food chain. While we can claim that house cats are killing birds and small animals, we also need to remember that in the same locations where cats now live, there used to be larger predators, such as wolves, that are no longer present. People who are responsible pet owners can consider keeping their cats indoors, where they won't be able to hunt; though the cats should be given plenty of playtime and cat furniture to make up for it.
I believe feline populations are sparse enough that they do not pose a serious threat to biodiversity on Earth. I think their numbers would have to be far higher in order for them to be a serious threat. That's not to say that they couldn't be responsible for the loss of some bird species.
No, felines do not pose a serious threat to biodiversity on Earth, because there is not an overpopulation of them. The ones that do exist are relatively harmless. They kill mice, but we do not have to worry about the mouse going extinct any time soon. Cats are something to have for a pet, or to cut up in biology class.