Yes as you say during the 17th and 18th century: when the British (and Dutch, etc) tried to take the remains of the "newly discovered" world because the scramble for the Americas was well over and Spain had already won in the 16th century.
By the early 19th century the entire World was quiet clearly divided, (the Scramble for Africa would later solve the glitches) and there was no room for the British to establish new evolving outposts anywhere in the Americas. Reinforcing the already claimed Falkland Islands is as much as that outpost could evolve since from an oceanic point of view, in the 1800s, you didn't need to spend money developing a very sorry colony in Tierra del Fuego to actually control traffic through Magellan or Cape Horn if you had more easily protected islands already in your control next to it.
Patagonia wasn't an option since it was formally and incontestably already claimed by Spain, simply not settled.
Instead of an effort to create a Crown Colony in the promising fertile Southern Cone, the British continued discovering and settling Australia, a task I suppose daunting enough, and a clear insular antipode for Britain (no messy land borders)
Also instead of such an effort, it was preferred to liberate all of the old enemy's empire in the Americas for immigration, intermarriage with supposed elites, and TRADE. Notice eventually the Hispanic American Republics all included some freedom of religion and speech, which coupled with the ability to trade and invest in those countries, is all the British Empire needed to operate at maximum profit.
A french approach would have been to flood the continent with bureaucrats which is in fact what the Spanish did, and the reason they lost their Empire so easily.
How could the British remotely control the Strait of Magellan and so forth from the Falklands, given that they're at least 300 kilometres apart (taking at least a couple of days to cross, in those days)? Who or which power directly controlled the strait itself, along with Cape Horn - at least until the Chileans came along in 1843 with the foundation of what would become Punta Arenas?
And you say that the British didn't need to take over Spanish territory. And yet the British did capture Trinidad in 1797 (but then again, many French-speaking slave-owning settlers - especially from Haiti, Martinique, etc. - were brought into that island by the Spaniards), and they attempted to capture Puerto Rico that same year, and Santa Cruz de Tenerife (on the Canary Islands) was also the subject of an attempted British capture in that year (by Horatio Nelson, no less). Furthermore, the British captured Havana and Manila in 1762 (only to give them back to Spain the following year). The British, moreover, captured Porto Bello, Panama, in 1738 - but in that case, they were after trading; thus, while Porto Bello as a trading centre was destroyed, that area was given back to Spain after just three weeks.
I think that Uruguay and Patagonia, among all the Spanish-speaking areas in South America, were the most ripe for British colonization/settlement in the 19th century. Uruguay was located along the frontier between the Spanish and Portuguese empires in South America, and it was fought between the Spanish and the Portuguese (later, the Argentines and Brazilians); even in real life, it was the British who established an independent buffer state in that zone, but that was well after Britain's experience in 1806-07 and the change of mind towards exclusively economic links at least largely as the result of the invasions' failure. As for Patagonia, you might be right that it was long under formal Spanish control and yet not settled; Belize (next to Guatemala and Mexico's Yucatan Peninsula) was officially under Spanish control, and yet such control was so ineffective that the English were able to establish settlements starting in the 1600s without any problem. Similarly, the British would probably have started settlements in Patagonia despite, at first, being under the official but ineffective control of Spain and its successor states like Argentina. For both Uruguay and Patagonia, there would also have been startegic considerations (in particular, with the Rio de la Plata and Montevideo harbour, and with the Strait of Magellan/Cape Horn) coming into play.