Actually, I think economics swung a bigger stick than Argentina ever could.
You have to look at England in the 1970's. Economically, they weren't in great shape. England had spent the last couple of decades dismantling its Empire. India, the cash cow of the entire thing, had gone in 1946.
After that you were left with Dominions like Canada, Australia, South Africa and New Zealand that were basically independent, or non-economic territories. Pretty much all of the Empire's African holdings were abandoned in the 1960's.
Through the 1970's, the British Empire withdrew from its remaining middle east holdings, Aden and Bahrain, as well as from the Pacific, Fiji and Vanuatu.
By 1980, all that was really left was Hong Kong, which was due to expire when the lease was up, and a few odds and ends, little far flung islands, many of them uninhabited or barely inhabited. Mostly of no economic or political significance, but damned expensive to maintain.
The Falkland Islands were in the 1970's and early 1980's were basically a giant white elephant. Literally on the other side of the planet, extraordinarily expensive. The entire population could be fit into a corner of a suburb in London, all it produced was sheep and the British could get wool cheaper and easier. The potential of offshore oil was unknown and decades away.
I think the British back in the day were actually looking for ways to get shed of the damned thing. The political problem was a local population which couldn't sustain independence on its own, and was looking pretty askance at the notion of joining up with Argentina.
Had Argentina made an agreement with Britain to submit its claim to international arbitration or mediation, and had it been able to offer credible guarantees of civil rights and economic support to soothe the Falklanders (or at least British consequences) I can tell you two things: 1) The British would not have fought their case very hard; 2) Upon loosing, the Brits would have been out the door before you heard it swing.
As it was, the Argentines saddled themselves with a psychopathically violent and vile regime whose evil was matched only bit its incompetence. That regime, having disgraced itself in the eyes of the civilized world then proceeded to go after the Falklands in the absolutely worst way and in a fashion calculated to inspire British presence for another 50 years.