I've seen many questions recently asking how Cortés managed to topple the mighty Aztec empire with his relatively small contingent of men and horses. The consensus seems to be that he didn't, and that his native allies did the brunt of the toppling. The Spaniards were merely the spark that set the fall of the Aztecs into motion, and they would have been wiped out had they not exploited the empire's internal divisions.
With that being established, how on earth did the Spanish consolidate their control over Mexico in the century or so after 1519? Why did their undefeated native allies meekly submit to the Spanish, who were merely one small part of a larger anti-Aztec coalition? How did the tiny number of Spaniards who settled there manage to establish full political control, place themselves at the apex of the Mexican social hierarchy, and herd the natives into encomiendas where they would live as serfs, adopting the Spanish language and Catholicism?
A similar question was asked a few days ago, you might be interested in the answers by myself and u/drylaw on the subject, might help you with your questions about the encomienda system and the first years of colonial politics over here:
https://www.reddit.com/r/AskHistorians/comments/gjz56j/how_did_the_spanish_manage_to_successfully/
Another related answer by u/611131 over [here] (https://www.reddit.com/r/AskHistorians/comments/gu0rib/how_was_hern%C3%A1n_cort%C3%A9s_able_to_get_so_many_native/) might also give you an idea of the forces at play during and after the Fall of Tenochtitlan.
An important addendum that I would make though is that Cortez didnt consolidate power in Mexico (the region) after the conquest, but rather in Mexico (the city), in fact spanish power in Mexico (and America in general) wouldnt be consolidated for over 200 years after the Fall of Tenochtitlan, many, many times the colony was at risk of collapse or outright conquest such as the defeat during the Mixton war or the many many times they were repeled from the Mayan regions, even after the "final" mayan polity of Nojpeten fell all the way in 1696 (more than a 100 years after Tenochtitlan), serious and succesful revolts would happen time and time again such as the Yanga rebellion (fair enough that was a slave revolt but you get the example) or the Pueblo rebellion, several Mayan in the south and others in the north like the Yaqui would continue to be important military factors well into the Mexican Republic... hell more than half of New Spain's population didnt even spoke spanish at the time of the War of Independence, according to Spain's own numbers, all the way into 1810, and I would bet that some of those that did, probably did so as a secondary language, not a maternal one.
Personally I would say that it was the Mexican Republic which consolidated power over Mexico, while the colony of New Spain only ever managed to consolidate around urban centers with anything outside those limits being well outside their control (though not their influence).