Okay, I'll take the biologists' word on it that domestic cats in North America qualify as "invasive"! Still seems to me (as a layperson) that after hundreds of years for an equilibrium to be established that they would no longer be considered invasive, but I'll take your word for it.
Right. But the 'equilibrium' is merely numerical, not natural. Human altered landscapes, and the fact that humans domesticated cats and tend to be tolerant of or encourage their presence (unlike many humans' attitudes toward larger native felid species), allows the domestic cat populations to persist in most places at MUCH higher numbers than any native felid can or could.
I will simplify, but broadly speaking: Under most natural, un-human-altered conditions, predators cannot exert a permanently depressive effect on their prey populations...because a healthy prey population is by definition occupying habitat that allows it enough food, water, and breeding and escape cover for enough individuals to escape predators and reproduce, and produce of 'surplus' of individuals upon which the predator can subsist. Populations thus naturally fluctuate around the carrying capacity of their habitat, and don't go into 'booms' or 'busts' unless some additional forces are at play.
But in the case of domestic cats, humans support artificially high populations by tolerating ferals, keeping cats as pets, breeding cats, keeping cats on farms, etc. Humans also alter the environment in ways that often further advantage the cats (most cities tend to be less hospitable to the larger predators that might kill the cats, and they support huge populations of other species that serve as prey and allow cats to maximize reproduction, eg, doves, grackles, and a bunch of OTHER invasive nonnatives: house sparrows, black rats, house mice, starlings, etc). In rural areas, agriculture simultaneously destroys native wildlife habitat AND supports artificially high populations of rodents (hence the farmers' need for the cats). So you get a disproportionately huge effect of a single predator species on all of the native prey species. In such a situation, a predator population CAN serve to permanently reduce carrying capacity of the prey's habitat and suppress prey numbers far below what is natural.