This is a facile thought experiment, and I'm surprised it's weakness hasn't already occurred to you.

But as it hasn't, let's change your island to an island of supremely skilled butchers. Your islanders are apparently capable of reassembling the mystery radios to restore function, so let's allow my island's butchers the skill of dismembering flesh atomically - and reassembling it exactly. Perhaps via quantum knives and very strong prescription lenses.

Their own religion has long prohibited them from peering into their tribe's flesh - and their tribe is all there is on that remotest of islands. But the day a missionary washes up on shore they see an opportunity. They can finally take a proper look. They begin with his head - by far the most interesting part of him, with the delighted and friendly if slightly smug smile and neverending chatter.

Once dismantled and laid out in trillions of tiny pieces in a large hall - a bit like a crashed plane - they decide to reassemble his head in different configurations - to test how the behaviour would change with each layout. There's an unusual fruit on the island which allows them almost superhuman focus and speed - so that billions of reconfigurations can be completed over an afternoon.

BUT WHAT DO YOU KNOW...? Every single reconfiguration results in a totally inert missionary! No more drunk on the milk of paradise look, and no more "Hallelujah!"s. Just a lump of meat. Perplexed - they decide there must have been something truly unique about the missionary's head's original layout and composition - so reassemble him in the way they found him. But when they have done this - calamity! - he doesn't come on any more. They check their work 18 times - but not a single atom is dislocated. And so the wise islanders come to the only conclusion possible, and hurry to note it in their scientific literature:

"Radios aren't animals. And anyone pretending otherwise is a ginormous silly billy."