"I don't think there was ever mass migration by the Spanish and Portuguese to South America."

This is not correct. There was mass Portuguese migration to Brazil, and mass Spanish migration to a few countries (such as Cuba, Argentina, and Uruguay). There was also significant Italian and German immigration to places like Brazil, Argentina, and Uruguay. Even in countries where there wasn't mass Spanish migration, the Spanish still made significant contributions to the modern gene pool. Genetic studies indicate that Mexico, for example, is split pretty evenly between European and indigenous ancestry.

"whereas in Latin America the indigenous populations were instead subjugated, rather than wiped out, and later those same groups overthrew the colonial leadership and established self-rule."

This is also a generalization that is not all that accurate. In some Spanish colonies, the native peoples were wiped out as much, almost as much, or even worse than in the US, Canada, Australia, etc. There are no full-blooded indigenous people left in Cuba, Puerto Rico, or the DR, and the percentage in Brazil, Uruguay, and Argentina isn't much higher than it is in Anglo colonies I mentioned (and in the case of New Zealand, the Maori population is in the double digits percentage wise). This characterization ignores the migration of people from Europe as well as the importation of African slaves, and the race mixing that occurred during the 300+ years of colonization. Furthermore, many of the revolutions that occurred in Latin America were led or heavily influenced by the criollos (people born in Latin America of full Spanish ancestry). This was in large part a result of the criollos having inferior status to peninsulares (people born in Spain).

"And I would imagine those countries wouldn't identify themselves as "Western", the same way Algeria didn't and does not identify as "French" despite being a part of France for centuries."

Algeria was a part of France for a fraction of the time Latin American countries were Spanish/Portuguese colonies.

"If you wanted to, you could certainly broaden the umbrella of "Western Civilization" to include all of the former colonial holdings in Africa, Asia, and the Americas, but then the "Western" label becomes somewhat pointless as a descriptor."

Not all colonies are alike. On average, indigenous influences are far greater and European influences far weaker in African and Asian former colonies than they are in Latin America.