Although domestic cats aren't native as such, they've been part of the ecosystem for many years, as far as I'm aware, wild bird numbers aren't declining significantly, and what decline there is appears to be more due to environmental and human factors. Which seems to suggest that there is somewhat a balance between birds born and birds taken, which is a very natural situation, regardless of our interference.
When otters were 'rewilded' to bring numbers back to their turn of the century numbers, they had a terrible impact on both wild and domesticated animals. The ecosystems were just not there to support such a prolific predator. That situation is somewhat balancing out now, but there are still numerous issues, that are far greater than domestic cats' impact on bird numbers.
However we feel about nasty cats killing lovely birds, and natural versus human-made effects, the basic nature of predators and prey would be happening regardless - sometimes there's a balance, sometimes it tips too far one way - too many prey animals isn't always beneficial either - and in a country that barely has any truly natural environments left, any kind of natural balance is long gone.
Whether due to rewilding or pets, or space for crops, or animals for food, there is no perfect (eco)system; farming vegetable crops kills thousands of small mammals and birds every year.
It's not uncaring, or cruel, or entitled to have uncollared outdoor cats, our behaviours as humans keeping predators as pets over decades, has shaped the wildlife we see as natural.