I can't equate the ordinary Argentinian with what their government is spouting. I went to BA about 5 years ago on business, and found it very agreeable. I thought I'd be ostracized, but far from it. The place itself is like a European capital, the people speak Spanish with an Italian accent, and half of them have British names. I was made very welcome, and had a good trip, made all the better by the locally produced Warsteiner, which is a very passable imitation of the real thing.
I agree the best thing for the UK is to say and do nothing. It should just die a death. In the meantime, regularly ask the Falklanders to confirm in a referendum which government/sovereignty they would prefer. Say every 4 or 5 years. Trot that out as ammo at the UN. Politically, job done. I just don't see a military solution on their part.