In the 13 Colonies, it took many years and a massive influx of colonists for the English to assert control over even the tiniest sliver of American land, and that only happened once white settlers came to outnumber the natives. In contrast, the Spanish colonies had relatively few white settlers spread out over a far larger and more populous area. It feels almost trivially easy for a local strongman to take up arms, raise a relatively large army, and overrun vast swathes of countryside while the white settlers cower in the major cities while it can take months for any semblance of organized response to arrive, by which point it's already too late and the rebellion had spiraled out of any ability for the puny colonial militia to put down, necessitating an expensive and unpopular response from metropolitan Spain. How did Spain manage to keep the remote American countryside from slipping out of its fingers?
In terms of colonial Mexico, the Spaniards did exercise control by using existing local structures and placing their own allies (at first), adapting the encomienda, or creating the presidio-mission system.
Many of the indigenous groups were dominated or subjugated by the Aztecs, so the initial Spanish arrival was seen as a way to "escape", alliances offered and the like, but actions like Cortes' in Cholula in October 1519 should have dispelled that idea. During the 16th Century centers like Veracruz, San Luis Potosí, Monterrey, Puebla, Acapulco, Durango, and Oaxaca were founded or transformed, creating outposts - almost spokes on a wheel centered on Mexico City.
The feudal encomienda system - forced / complelled service or tribute offered to the crown by proxy by other nobles/elites - was imported and modified that resulted in forced labor by indigenous groups to large landed estate owners. While the encomenderos were to educate them in Spanish, the faith, and such, it turned into a coercive, brutal, exploitative system. Fray Bartolomé de las Casas, OP wrote extensively about this practice and managed to the Spanish Crown to reform the practice in favor of self-governing "Indian" townships, but what was decreed in Spanish wasn't really implemented in New Spain. His Entire los remedios (1552) and Brevísima relación de la destrucción de las Indias (1542) were instrumental in this effort.
Several of the religious orders, especially the Franciscans and Jesuits, specifically had higher level schools (Colegio de Santa Cruz de Tlatelolco [Franciscans] and Colegio de Mexico [Jesuits]) to educate in European fashion the sons of the elite, such that many were fluent in Nahuatl, Latin, and Spanish and oriented to European thought and Catholicism, some of whom were ordained priests. Cushner's Soldiers of God: The Jesuits in Colonial America, 1565-1767 is a good source.
In the northern areas of New Spain, either loyal groups were offered lands and relocated from central Mexico or further afield the presidio - mission system was used to create reducciones where a presidio was placed to support the activities of several missions that attempted to have indigenous groups settle near the missions and abandon the hunter-gather lifestyle while being evangelized. The missions of CA, AZ, NM, and TX are examples.
However, on the fringes of the colonial area, control offered by the mission-presidio system could be tenuous, at best. Francis Galan's *Los Adaes, the First Capital of Spanish Texas (*College Station: Texas A&M UP, 2020) adresses the failed efforts of Spain to bring this contested of area of New Spain (modern day TX-LA border) into their economic and political system.