Cats survived before us by hunting small mammals and small birds, and they are very effective at getting fed.
And, conversely, the prey evolved to avoid cats. So it is only a problem if you take cats to a place that historically did not have them. In fact, removing a predator from an ecosystem it used to keep under check can be just as devastating as introducing a foreign species.
Literally nowhere historically has had cats. Wild cats existed in Northern Africa/Mediterranean regions about 10 to 15 thousand years ago and were from there spread by human agricultural revolution to be introduced throughout Egypt, Rome, and then Roman Colonies as well as Asia, and some thousands of years later they exist on every continent except Antarctica.
The tiny speck of area and population that they should naturally have is like a grain of sand on a beach compared to the destructive force they have become.
Literally nowhere? What absolute bollocks.
Cats moved across a land bridge to the UK over 9000 years ago, long before the Romans had anything to do with it.
F. Silvestris, the European Wildcat, is generally considered a separate lineage from domesticated cats, though somewhat capable of crossbreeding, and because of human introduction of domestic cats the Scottish Wildcat in particular is functionally extinct in the wild. Just one of many great examples of the destructive nature of this pet and human negligence.
As you yourself said, cats have been living across most of Africa, Asia and Europe for over a thousand years. So unless you are talking about Australia, the Americas, or a few corners of the old world, cats are either native or naturalised enough that they are now a part of the ecosystem.
A thousand years is nothing to an ecosystem. Birds have been migrating across Europe, Asia, and the Americas for hundreds of millions of years, only to get slaughtered in droves by furry shit machines.