I just encountered this old myth again on another blog. My answer?
Nothing is natural about an alien introduced pest preying on native species! Here's a bit of wisdom in a poem written many long years ago. It applies exactly to our problem with feral cats in our ecology today.
She had a name among the children;
But no one loved though someone owned
Her, locked her out of doors at bedtime
And had her kittens duly drowned.
In Spring, nevertheless, this cat
Ate blackbirds, thrushes, nightingales,
And birds of bright voice and plume and flight,
As well as scraps from neighbours’ pails.
I loathed and hated her for this;
One speckle on a thrush’s breast
Was worth a million such; and yet
She lived long, till God gave her rest.
Born 1878, killed in the battle of Arras- Easter Monday, 9 April 1917