As a Briton myself, I don't quite understand why you find this a ridiculous argument. I mean, the Falklands' nearest neighbor is Argentina, but they are are 13,000 kilometers from Britain.
Of course, it's the inhabitants who are the only ones who matter in the dispute, and they voted overwhelmingly to remain, making them smarter than your average Pom(*) was in 2016 anyway.
I liked that section, but you missed one key detail - how tiny and, ah, what's a polite word for "useless"?, those islands are.
There are just over 3000 people there, and no particular natural resources - the economy runs on fishing, sheep and tourism.
And yet this tiny island with so little to offer has been a source of dispute for two centuries!
Humans. So silly.
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(* - Pom or pommie is Australian slang for a British person - it comes from Prisoner of Her Majesty. :-D)