1. The islanders have asserted their desire to be British citizens.
2. The UK has held claim to the island for nearly 200 years (far longer than any claim Argentina may have)
...as a matter of fact, what claim does Argentina have to the island other than proximity?
The Commonwealth of Nations has 135 million citizens across the world. Why are the Falklands any different than New Zealand?
Whether or not they can still be protected is simply conjecture on your part, and not relevant to sovereignty of the islands.
The Irish asserted their desire to kick the Brits the hell out of Northern Ireland. Did the Brits respect that?