The true picture of this gets confused by the dominant narratives, which, as you say, are that Latin America was conquered quickly by the Spanish and Portuguese, while the USA was gradually settled over hundreds of years. In reality, while the initial conditions were different in part (i.e. more complex civilisations in Latin America), the conquest and settlement of Latin America was also a hundreds of years long process just like that of the USA was.

Of course as you mention, there was one big difference, which was that in Latin America there were several civilisations, of which the Aztec and the Inca were the most prominent, which were far more organisationally complex and larger scale than those in the area of what became the USA. Spanish conquistadors conquered the Aztec and Inca empires quickly in the early 16th century, and put themselves at the top of the existing social pyramid, replacing the indigenous elites. So in this way indeed these territories suddenly became part of the Spanish empire, because there was already this complex social organisation in place with which to rule. Think of something like the Norman conquest of England in 1066 for an analogy, a very small number of men defeated the army of the King of England, and therefore became rulers of England themselves, because there was already this ruling structure in place. Replacing the ruling elite, not mass settlement of people, was the norm of how conquest happened in the pre-modern era. In the present day USA was these kind of societies (complex enough that Europeans could put themselves at the head and become a land owning rent seeking aristocracy) didn't exist. As you may know, Hernando de Soto did explore the USA for this purpose, but didn't find what he was looking for.

However even given this difference, there's still more similarities between the European conquest and settlement of Latin America and that of the USA than people sometimes think. It's easy to be deceived by looking at maps of the Spanish empire in the 16th century, or by thinking something like the Aztec empire = Mexico today, to believe that within 50 years or so of Columbus's arrival, Latin America was all ruled by European powers. But google for example for a map of the actual size of the Aztec empire, it's a small fraction of present day Mexico. The Inca empire is depicted as being somewhat bigger, but is still only a small part of the continent. A more accurate representation of European colonisation in Latin America would be similar to the time lapse ones of the USA, with small pockets of settlement (though starting from a larger base due to the complex Aztec / Inca ruling systems) gradually moving inland over a period of hundreds of years. For example it took 100 years for the Spanish to conquer what is today the northern Mexican state of Chihuahua, and even into the 19th century, the northern areas of Mexico were barely under government control (that's why they invited the Anglo settlers into Texas). In Yucatan, the Maya rebelled in the mid 19th century and preserved their own independent state until the 1890s. Argentina didn't conquer and settle the vast majority of its territory until the 19th century ('conquest of the desert), for hundreds of years before this the territory had pretty much just been the area around the Rio de la Plata. Chile went through a similar process (Occupation of Araucanía). In Brazil settlement was limited to the coasts until the late 17th century, only then did they start pushing inland, and they still are today! (e.g. uncontacted tribes in the Amazon).

So in fact aside from the initial conquests of complex civilisations, the European conquest and settlement of Latin America was quite similar to that of the USA, starting small on the coasts, slowly over hundreds of years extending their rule, until the 19th century, with its technological and population explosion, saw a rapid increase in speed of European expansion until they truly controlled the inland.