Then the naturally-occurring flora and fauna assemblages on New Caledonia got lucky, if your anecdotal observation is to be believed.  Without substantiating data I doubt it.

Cat lovers tell me all the time their little fluffies don't harm birds when they let them out.  That's of course untrue unless fluffy is blind, deaf, crippled, clawless and toothless.  And if infected with toxoplasmosis fluffy still kills wild mammals and birds by way of its contaminated feces.

Insular ecological communities are particularly vulnerable to human-engendered introductions.  Hawaii lost half its native birds to introduced cats, mongooses, pigs, dogs, rats and mosquitoes.  New Zealand lost 40% of their native bird species to cats, ferrets and rats.  Guam has lost 11 bird species to the invasive brown tree snake.

Invasive Indian Mynahs, European starlings, English sparrows, chickens and even great-horned owls have been implicated in native bird declines and extinctions.  But turkeys are OK!  Right.