Well the islands are settled now and they're settled by people who think they're British so I can't really see how you can functionally come to the conclusion that there is any mystery.

Argentina's claim to the islands seems to be their they are sort of close to Argentina, although far enough out to be an international waters and there was once a Spanish settlement there back before Argentina existed as a country. By the time Argentina became its own country the Spanish settlement was no longer in existence.

So from most of human history the islands were uninhabited or only sporadically inhabited and we're talking the level of driftwood huts here, no permanent structures. If anyone had any legitimate claim to the island it would be Spain they were the first to build proper buildings on the islands, at least they had a settlement there until they abandoned it, but then they quite happily sold the rights to the land to the British.