France was the first to arrive, and they give up the islands to Spain, UK arrived a year later claiming "no knowing" the existance of the France settlement.
Later Spain agreed to let UK stay in the islands for a few years, but Spain will retain sovereignty(that was to avoid a war), later, as you say, they left because of the problems in America, but they leaved a sign claiming the islands as British territory...
They came back in 1833, expelled the population and bring in their own.
How someone may think that the islands belong to UK or the pepolation has any right to self determine is beyond me, this a case driven by interests, anyone that supports UK its because is interests, very few are intereseted in support Argentina claims.