The UK Foreign and Commonwealth Office (FCO) regarded the islands as a nuisance and barrier to UK trade in South America, so, whilst confident of British sovereignty, was prepared to cede the islands to Argentina. When news of a proposed transfer broke in 1968, elements sympathetic with the plight of the islanders were able to organise an effective Parliamentary lobby to frustrate the FCO plans. Negotiations continued but in general failed to make meaningful progress; the islanders steadfastly refused to consider Argentine sovereignty on one side, whilst Argentina would not compromise over sovereignty on the other. The FCO then sought to make the islands dependent on Argentina, hoping this would make the islanders more amenable to Argentine sovereignty. A Communications Agreement signed in 1971 created an airlink and later YPF, the Argentine oil company, was given a monopoly in the islands.

In 1980, a new Minister of State for Foreign Affairs, Nicholas Ridley, went to the Falklands trying to sell the islanders the benefits of a leaseback scheme, which met with strong opposition from the islanders. On returning to London in December 1980 he reported to parliament but was viciously attacked at what was seen as a sellout. (It was unlikely that leaseback could have succeeded since the British had sought a long-term lease of 99 years, whilst Argentina was pressing for a much shorter period of only 10 years.)

Where's the unwillingness by the British to engage diplomatically? It seems like their FCO was fine giving the islands to Argentina, but the biggest obstacle was the people living on the islands. Britain tried at least three approaches: a straight transfer, a closer commercial connection of the islands to the mainland, and finally a Hong Kong-style lease scheme. It's honestly hard to argue they could have done more -- they tried to just give them away right off the bat! The only country acting outside of multinational organizations here was Argentina, when they abandoned diplomacy and invaded.

But maybe we should mark “appeasement” for discussion in the next session. I think it really doesn’t hold in this case

The junta had launched a quickly-aborted invasion of Chile in 1978. There are also oil interests that had already spawned one war near Argentina's border, and within the territory that had originally broken away from Spain as one unified country (before Bolivia and Uruguay split off). There were ample opportunities for continued aggression.