Britain beat Argentina in terms of discovering the islands and establishing a presence there. Yes, Argentina had a brief presence there too. But it was only after Britain had become involved in the islands.
This is arguable. Firstly, there is no agreement on the discovery of the islands. The usual British claim is that they were discovered by John Davis in 1592, followed by Richard Hawkins, but the Spanish & Portuguese both dispute this, claiming earlier sightings - and since they'd been sailing around those waters for over 70 years by then, it's quite likely that they're right. There were several documented sightings of & some visits to land that could have been the Falklands before 1592. There was even an English one - Drake. It is also disputed whether the land Davis & Hawkins saw was actually the Falklands.
The first completely undisputed sighting was by a Dutch navigator in 1600.
Secondly, we were not the first to establish a presence. That was done by the French, in February 1764. We followed in 1766 (there was a visit in January 1765, 11 months after the French settlement, but no settlers until the following year), not knowing the French had beaten us to it. The Spanish, however, had spotted the French settlement, objected to it, & got recognition by the French of their priority, all before we noticed the French presence in December 1766. In April 1767, the French formally handed over their settlement to the Spanish. Usual rules would have that dated from February 1764 - continuity of occupation, recognition of title, etc.
Our presence in the islands was relatively brief: 1766 to 1776, when we voluntarily withdrew, & with a hiatus in 1770-1 when a Spanish flotilla forced (without fighting) our temporary evacuation. We retained a formal claim, but did nothing about it for 53 years, & nothing except verbal protests for 56 years.
Argentina claims (& this is normal, & generally accepted in such cases) that it is the heir to Spanish jurisdiction. The Spanish settlement, run from Buenos Aires, was occupied from 1767 (1764 counting the ceded French foundation) until 1811. By then Spain had lost control of what is now Argentina, & the settlement (without a Spanish governor since 1806, the Spanish having urgent business in Europe) was evacuated in 1811 by the provisional revolutionary government of Rio de La Plata, which was to become formally independent as Argentina.
From 1811, there was no government in the islands for a while, though there was occupation, by a motley international crew of sealers & whalers. The first attempt to establish a government of sorts was undoubtedly Argentinean. There was a formal claim & flag-raising in 1820, the purely nominal appointment of an absentee governor in 1824, & more significantly, its recognition of Luis Vernet as having delegated authority, & his colony, established gradually from 1824 onwards. Vernet's colony was pretty well wrecked by the US navy in 1831, & Argentina responded by sending a governor, & when he was murdered, an Argentinean navy officer took charge.
When the Royal Navy arrived in 1832 (our first official presence since 1776), it found an Argentinean warship, & a naval officer trying to impose some order with the aid of his crew. The Argentineans most definitely did not establish a presence only after we'd got involved. Our involvement was in response to the Argentinean presence. It took several years of their presence before we even complained, & three years after that before we did anything more.
We have a perfectly good claim based on 179 years of occupation & government, over 170 years of settlement, & the self-determination of the population. We don't need to pretend we were there first.
