But nobody lived there except some gauchos and soldiers (many of whom were British)
I mean, that's blatantly not true.
From the wiki article ...
France was the first country to establish a permanent settlement in the Falkland Islands, with the foundation of Port-Saint-Louis on East Falkland by French explorer Louis Antoine de Bougainville in 1764.[2] The French colony consisted of a small fort and some settlements with a population of around 250.
A pop of 250 is not "some gauchos and soldiers". They were not even "(many of whom were British)".
I mean, we can go down the rabbit hole and start a population census conversation based on year-to-year, but that seems excessive for the conversation being had, and something that is really not needed.
Its fair to say that the French had a presence there, they gave that presence to Spain, and Argentina inherited that presence from Spain (going around the long way, as the Doctor would say).