I think some people need to appreciate just what is at stake. We're talking about an entire group of people losing its territory.
Handing over the islands to Argentina would achieve nothing except the bolstering of Argentine nationalistic pride. It would mean that an entire group of people would have to face losing their home islands, on which most can trace back ancestry over many generations, as they know them. Assuming Argentina doesn't intend on occupying the territory against the will of a hostile population, the inhabitants would either be forced to flee or remain on the islands as the fabric of their homeland changes from the one they know to a Spanish speaking, Argentinian one, with gradual large-scale immigration of Argentinian citizens.