This sort of thing is a huge problem here in New Zealand. The only native mammal here is a bat, we have mostly birds which evolved for a really long time with only avian predators. So they’re hilariously poorly adapted for surviving standard predators (cats, rats, dogs etc) which first the Maori and subsequently Europeans brought. For example, many of them are flightless and tend to freeze when threatened - works well against eagles but is a terrible idea when threatened by a cat.

As a result, we have many animals, mostly birds, which are totally unique and also critically endangered. Many of them can only survive on offshore islands which have been comprehensively cleared of predators at vast effort and expense. The islands need to be relatively accessible since humans have to get to them to maintain them, but it turns out that once in a while a predator will swim quite vast distances for no apparent reason, and it only takes one to mess up years of painstaking work. Quite apart from killing a bunch of birds whose total remaining numbers might range from the tens to the hundreds of individuals.

Nit: If predators periodically make their way to the islands without human assistance, don't the islands have native predators, by virtue of how we've woven ourselves into the definitions?

I love the visual of humans desperately trying to preserve what they consider the natural world, and when they turn their backs evolution does it's thing.

I too am flightless and freeze while threatened

Alcatraz isn't really that far from land, about a mile away. They have events where you can swim to and from it. The currents make it dangerous, but the distance is unremarkable.

Direct ecological management is unfortunately a bit of a game of using a bucket to fix a leaky ship. The equilibrium that established the ecosystem dynamics in the first place is disrupted. A new equilibrium might form over time, but we enforce the old one because that is what we documented when we first came to a place, even though it is no longer thermodynamically favorable.

Ironically, the ecology of an island itself came from events like a random animal swimming to it over the historical record and finding sufficient spare resources or an ecological niche they could satisfy sufficiently to reproduce. Distance from mainland and species diversity is very strongly correlated reflecting increasingly scarce odds of these "heroic journeys" at greater distances. Species themselves are capable of exhausting an islands resources and putting themselves into local extinction even with no human intervention (such as the case of the last of the mammoths on wrangel island).