I'm afraid this is inaccurate.
Argentinas main claim is historical, with geographical proximity being a minor factor. The islands were claimed by Spain from the early 17th century (& possibly earlier, if the islands sighted & claimed, but not accurately located, from Spanish ships in the early 16th century were the Falklands), & governed by Spain from the 1760s. The first settlement, & hence effective claim, was French (BTW, the Spanish name is a Spanish version of the French name - Les Malouines, from St. Malo) - and subsequently ceded to Spain, & the Spanish claim recognised by France. The Spanish withdrew their garrison when wars elsewhere needed the troops, & governed the islands from Buenos Aires (with regular official visits to oversee the itinerant whalers & sealers there) until Argentinas war of independence.
Argentina has claimed since becoming independent that it inherited the islands from Spain at independence. It attempted to enforce its claim from about 1819 until forcibly ejected by the UK in the 1830s.
The first British attempt to settle the islands was a couple of years after the French settlement - 1766, IIRC. After a few years squabbling with the French & then Spanish, the settlement was abandoned in the early 1770s. There was no further British attempt to occupy the islands until 1834, i.e. 18 years after Argentinean independence. It is therefore wrong to say that the islands were part of the UK, or even a British possession, before Argentina existed. BTW, they never have been part of the UK. They're in a different category, & always have been.
Spain holds that Britain recognised the Spanish claim in the Treaty of Utrecht (1713). The United Kingdom disagrees with that interpretation of the treaty.
The main (IMO only) strength of the British claim is based on 176 years of uninterrupted possession, 169 years of settlement, & the wishes of the inhabitants. That's a pretty solid basis, & I believe perfectly adequate. There is no need to embellish it, & IMO making false claims about 'part of the UK', 'settled for 300 years' (not you, I know, but I've seen it) undermine, rather than strengthen, our claim.
BTW, there's one little legal oddity. The first purchase of land in the Falklands was made by the precursor of the Falklands Islands Company from a man called Louis Vernet, who was then (1840) visiting London. Vernet owned the land because it had been granted to him in the 1820s by the United Provinces (now Argentina). The British government recognised the validity of that land sale, which implies that it accepted the validity of the Argentinean land grant.
Oops!