Furthermore, in the 17th and especially 18th and early 19th centuries, tthe British established trading posts in India, Malaya, and parts of West Africa that eventually evolved into much larger British colonies. So, too, at least 1-2 British trading posts in South America could have led to small(ish) British colonies amidst all the newly-independent countries.
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.