This may be way out of the scope of this sub, but why are the only non-European nations that are considered a part of Western civilization the former British colonies? Latin America shares with "Western nations" a great deal of history, religion, and often politics and race (Argentina in particular is largely made up of people with European heritage). These days they're largely democratic and increasingly wealthy (with Chile a member of the OECD).
The definition of "the West" grew out of the definition of Greek "western civilization" and later Western Europe, and the definition continues to expand in modern times to be more of a formal/informal political/economic/cultural bloc. Places like the US, Canada, Australia, and New Zealand are often associated with "the West" because of the their last century of entanglement with Western European Powers via involvement in the World Wars, NATO, etc. Unlike the US, Canada, etc. Latin American nations largely sat out of the world wars, and were not really part of the official "western bloc" facing off against the "Eastern bloc" in the post-war. It's for these reasons that places like Russia are not considered part of "the West", but former Eastern states that have subsequently joined NATO are. Conclusion, the term "the West" is mostly a political definition that seeks to define a culture and alliance, but it is largely fluid, and not necessarily a geographical. Others theorists defining western civilization as an outgrowth of Christianity and European settlement, rather than a modern political definition, would include Latin America and the Caribbean as part of the Western World.
As a point to begin with, your question is entirely based on the idea of Latin America as "excluded" from the Western nations, rather than actively following an independent path (which especially in regards to the Cold War development of the 'Third World' is pretty important). Not a criticism, but try and make sure you're approaching it from different angles!
On the former British colonies vs the Latin American colonies, an important point is that it is only those former British colonies which were seen as white settler colonies which are regarded as the "Western nations" - Canada, Australia, New Zealand and South Africa - those countries which were first granted a degree of self-governance and eventually Dominion status.
That model of decolonization was an entirely different dynamic from the Latin American independence struggle of colonial elites against the Spanish empire, and often a different type of initial colonization as well. Also the British Empire was a huge part of the construction of the idea of the "Western nations" and acted to integrate its Dominion colonies into that culture, whereas the Latin American countries were forging a separate path for a much longer time.
Specifically, I can only really speak with a lot of knowledge about Chile. In that case, race and the Cold War are important.
Genetically, most of Chile's population is highly mixed between the original Spanish invaders and native peoples, most notably the Mapuche. There was certainly a different attitude between that population and later 18th and 19th century colonizers such as the German and the British who tended to see themselves as separate and superior to the general Chilean population. Meanwhile leading up to the present, there is a strain of Chilean politics and society that takes pride in that Mapuche heritage and uses it to differentiate itself.
Politically, the Cold War carries huge importance. Salvador Allende was regarded as one of the leaders of the Non-Aligned movement and as the first democratically elected socialist President, clearly not part of the "West" as defined against the Soviet Union and Communism. After the coup of General Pinochet, Chile then adopted the neo-liberal economic model but clearly not the liberal democratic structure that at least rhetorically is used to define the West.
Your final question probably strays too much into present-day history for the subreddit rules but I can at least offer some anecdotal evidence that is more than 20 years old! During the 1980s, a rock band called 'Los Prisioneros' was very popular and often criticized the dictatorship and other parts of Chilean society. Their song 'Por qué no se van?' is all about criticizing Chileans who want to be "second-hand Westerners", taking all their cultural inspiration from the US and Western Europe. So clearly there was part of Chilean society that did aspire to that cultural world, but many other Chileans who refused to accept that.
hi, you'll find more info in these threads
Why is Latin America not considered Western? - a quick answer by /u/mystical-me
[META] Latin American History: Forgotten by the West? - a round-table discussion featuring /u/Irishfafnir and /u/whitesock among others
When I was studying political science "Western" was often a carry-over from the Cold War division between East & West, First World Countries & Second World Countries (with the unaligned states being the Third World).