The British didn’t discover the islands - the French (and then the Spanish) did. But they won them after that, so your points are still good - just wanted to point this part out

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.

I agree with this take. The claims are plausible from both sides but England obviously won the battle. The inhabitants being established are a result of the military positioning, so their votes will obviously be skewed. Not sure their votes are comparable to those of say native populations of other disputed territories (since there were none).

So I can understand why the Argentines feel aggrieved.

Hm that's tough. I always like to think boundaries set by countries are bullshit anyway, but it is located closer to Argentina. Another comment says they both want it because the island comes with vast amount of fishing so economic reasons. And the UK probably put money forth to develop some sort of living situation on there.