Domestic cats’ population is orders of magnitude larger than the native cats population was or is.
We've demolished the ecosystem of the native cat, so its no surprise they're at the brink of extinction in the modern day. I haven't seen any numbers on wild cats prior to the arrival of European settlers. But given the lack of demographic analysis, I'm betting these figures have enormous error bars, even assuming you're conjecture is correct.
Might not help that predatory birds and wolves have also been virtually wiped out. So domestic cats have no natural predators. But - again - that's not due to the cat population. It's due to the human population. Cats aren't the ones hunting wolves with sniper rifles out of helicopters.
the invasive species that is the domestic housecat is also talking a toll of a couple billion birds per year.
Habitat destruction has consistently been the primary cause of population decline, dating back to the extinction of the passenger pigeon. Housecats aren't killing birds at a meaningfully faster rate than prior era natural predators. Bird breeding is whats on the decline, with an enormous part of that decline coming from the decline in the insect and plant population that birds subsist on.
The fixation on cats is one more example of the personalization of an industrial extinction event. You're deflected from blaming pesticide manufacturers and real estate developers, so that you can scream your head off at your neighbor's SPCA rescue. Killing all the housecats will not bring back the bird population because housecats aren't the ones destroying the habitats birds need to hunt and breed.
You choosing to disbelieve them is about as valid as believing vaccinations are more harmful than diseases, that climate change is a hoax, or that illegal immigrants are responsible for the housing shortage.
Blaming cats for the declining bird population is exactly like blaming immigrants for the housing shortage. These are practically one-to-one comparisons.