As is the case with many conquests, the long-term success of the Spanish invasion of Mexico was able to occur because of how well the conquistadors were able to co-opt local nobles, right from the start.
The popular epic tale of Hernan Cortez begs the question of how a few hundred Spaniards conquered an empire of millions- and from Bernal Diaz de Castillo (one of Cortez's soldiers) to Paul Prescott (a prolific 19th century scholar) the superiority of Spanish arms, faith, and their incredible boldness was deemed a sufficient answer. Yet, the real key to victory was the coalition of indigenous peoples hostile to the Aztecs. The Spaniards were almost defeated by the independent Tlaxcala, before the Tlaxcala instead decided to make allies out of these strange, bearded men. As the war progressed, more indigenous groups, previously subjected by the Aztecs joined this coalition (Texcoco was likely the largest of these) and after a bloody siege, Tenochtitlan falls.
It may seem like I've gone off an a tangent here, but the point to all this is to explain why some local nobles were more likely to keep their status- those who helped the Spanish in the conquest held the most authority post-conquest. For example, groups of Tlaxcala warriors continued to follow various conquistadors as they spread to more distant parts of Mexico and Central America and often settled with the Spaniards as a foreign ruling elite over the local people.
This brings me to the next and likely most important point. The Spanish colonial empire was divided in two parts: the 'republica de españoles' and the 'republica de indios'. This meant that, in each settlement, there were essentially two distinct political entities, one composed of Spaniards and one composed of the local peoples, both of which had rights, privileges, and for the indigenous, labor requirements. Instead of labor, the Spaniards were expected to provide for military defense. The local nobles, so long as they remained loyal, thus could continue to lead their people, but now as subjects of the crown.
Of course, as the original question anticipates, there is intermingling between these groups. Spanish-born ladies were highly valued as marriage partners by the conquistadors and their descendants, but there were far too few of them. So they married the daughters of indigenous nobles (but it was exceedingly rare that a local man would marry a Spanish woman). This had the dual effect of providing a partner but also strengthening bonds with the neighboring 'pueblo de indios.' As in Europe, marriages were used to cement alliances in Pre-Columbian Mexico, but the difference was that most indigenous societies had no problem with polygamy- thus the Spanish nobles either weakened themselves locally by only having one marriage tie, or weakened themselves in the eyes of the Church by having multiple wives. Both situations occurred, at least initially.
Finally, a note on race, which was perceived in a completely different way than it is now. The division between the 'republica de indios' and 'republica de españoles' was not based on firm ethnic grounds, but rather on behavior. There came to exist a variety of terms for people of mixed ethnic background: meztizos, castizos, mulatos etc. but skin tone was far less important than dress, language, diet, and faith in creating a personal identity. Local nobles, or more likely the children of local nobles with Spaniards or even individuals of a less pure ancestry could participate the 'republica de españoles' if they could pass as 'Spanish,' on these cultural grounds. Or they could live in the 'republica de indios,' still potentially as a leader but of a distinct group, depending on their levels of assimilation.
My knowledge on this is mostly relegated to Mexico, so hopefully an Andean historian will come along and explain how it worked after the Inca empire fell.
Some sources: Seven Myths of the Spanish Conquest- Matthew Restall The Body of the Conquistador- Rebecca Earle Indian Conquistadors- Ed. Laura Matthew and Michel Oudijk
Although the surrender of Cuauhtemoc to Cortés following the Battle of Tenochtitlan in 1521 is seen, in retrospect, as the culmination of a grand clash of cultures and a pivotal turning point in history, we have to remember that, at the time, it was just another battle. A significant one even to its contemporaries, for certain, but devoid of the fraught connotations we now attach to it.
Instead, the next couple decades after the Conquest were, as Lockhart puts it, in a sort of stasis. The Spanish set up enclaves and ultimate authority was now vested in them via the Spanish Crown, but daily life proceeded without much interruption. This included rule of cities and towns by the same people who had been governing them before. Outside of the newly christened San Juan de Tenochtitlan, Spaniards were a very very tiny minority in those first few decades. Even in Tenochtitlan the Spanish mostly restricted their direct administration to their own enclave, with the rest of the city falling under the jurisdiction of the newly appointed cabildo.
The cabildos were essentially town councils made up of regidores (representatives) and alcaldes (judges/magistrates). Theoretically, these offices were open to election, but, particularly in the early days, they were filled by the same elites who had been part of the ruling class before the Conquest. Similarly, tlatoqueh of cities and towns quickly obtained the title of gobernador, which, at the start, was a lifetime appointment. Meet the new boss, same as the old.
The acquisition of official Spanish titles did not eliminate the Indigenous titles. Even before these colonial systems were set up, the Spanish Crown was already recognizing the authority of the "natural lords" of the common folk. These arrangements, and appeals to these arrangements, could be through writing or one of the several delegations of Native elites who traveled to Spain, most notably the group of a few dozen Mexica elites who traveled with Cortés to meet with Charles V in 1527.
For decades the two systems ran in parallel, which could cause some friction on the common folk (macehualtin). Often they had both taxes and labor burdens to be paid to the Spanish, and still had to fulfill tribute obligations to indigenous elites. Those elites themselves were being squeezed, as they also had taxes to pay to the Spanish Crown, even as they increasingly found their authority curtailed, their lands transferred to other (often Spanish) hands, and the population under their direct authority decreasing both through disease and individuals migrating to get away from their double-tribute burden.
Ironically, the assumption of Spanish colonial titles by the native elites preserved their authority temporarily, but ultimately curtailed and undermined it. Whereas before a powerful tlatoani could rely on a complex web of informal obligations to maintain authority over a wide swathe of other polities, they now had a formalized system of cabecero (head) and sujeto (subject) polities. In some sense this was merely a formalization of the relations which had previously existed, but it also allowed for a form of legal jockeying through appeals to the Spanish which had not existed before.
Gibson, in The Aztecs Under Spanish Rule, gives the example of Huexotla, which was a sujeto under the cabecero of Texcoco. Huexotla appealed this status using an argument rooted in pre-Hispanic rule, that the tlatoque of Huexotla had traditionally been independent from Texcoco, and thus Huexotla should be a sujeto in its own right. While Huexotla did not end up with full sujeto status, it was granted more independence from Texcoco as a result of this legal appeal.
Contrast this with what would have happened had the tlatoani of Huexotla opted to declare independence from Texcoco during Aztec rule. This act of open rebellion would have been met with military force and possibly even the replacement of the scofflaw tlatoani with military governor appointed by the Aztecs. The transfer of the monopoly on violence to the Spanish Crown, however, meant that previously subservient towns now had the ability to argue their case before Spanish judges.
Colonial titles undermined the traditional elites at lower levels of authority as well. It was eventually decreed that gobernadores would not be lifetime appointees, and the rule that alcaldes and regiodores could not be automatically re-elected to those positions once their terms were up was more consistently enforced. Combined with the demographic hits to the native population in the 1540s and 1570s, this meant colonial offices were increasingly held by men who were not tied to the previous system of nobility. Kellogg, in Law and the Transformation of Aztec Culture, notes that, by the 17th Century, a new class of elites had emerged, whose status was independent claiming descent from a tlatoani, but was explicitly rooted in their participation in the Spanish colonial system.
These new elites often had no interest in maintaining the traditional system of power, and could be hard-nosed and practical about how they intended to economically benefit from colonial office. Alfonso de Zorita, a Spanish judge sent to Mexico in the mid-16th Century, noted even then that:
The Indians who have replaced the natural lords in office are naught but a pack of thieves, for they are quite content to have the people bear their sufferings in silence. The new officials do not wish or seek the reform of abuses, for they are very well off under the present system.
To close things out, I just want to briefly give the example of the Don Juan de Guzmán Iztollinqui, who succeeded his brother as tlatoani of Coyoacan in 1526, as was recognized as the ruler of that town by the Spanish. He integrated well into the Spanish colonial system, fighting in wars of expansion and was granted a coat of arms and noble priviledges. Following the tradition of the indigenous elites in keeping their dynasties purely of noble native blood, he married a daughter of Texcocoan elites, which earned him even greater land-holdings than he already had.
Don Juan died in 1569 as a wealthy elite, his indigenous status recognized all his life and having served a gobernador of Coyoacan. His successors did not have such luck. His eldest son ruled only for a brief time before passing away in 1573, and the two younger sons who followed both perished in the huey cocolitzli epidemic of 1576. The succession crisis that followed saw a different family of native elites rising to prominence, and the next few decades were marked by power struggles which saw the authority of the office of tlatoani undermined and many of the traditional lands and rights redistributed, often to the new Spanish and mestizo elites. The widow of a grandson of Don Juan would eventually assert her legal rights to the holdings in 1604, but died without an heir, touching off an even murkier succession crisis which eventually resolved with another legal decision recognizing another Don Juan descendant who also died without an heir, although at this point there was not much left to be heir of. As late as the 1800s, the official officeholder was still trying to reclaim the lost lands, while supporting himself as a carpenter in Madrid.
The story of Coyoacan is a microcosm of what was happening throughout Mexico after the Conquest. The "natural lords" were initially recognized as such, but the colonial system opened up ways for non-elites to advance along a parallel path to wealth and authority. The old elites now had new competitors, in addition to distant relatives, cadet branches, mestizo cousins, and newly arrived Spanish all seeking to claim their own portion of the old elite's land, wealth, and authority. The fractiousness and explosion of contenders undermined the prestige of indigenous titles, even as epidemics undermined the populations which had supported those titles. The denigrated titles still had some weight, but the population of Mexico, which was increasingly Spanish and mestizo, found greater opportunity in working within the new colonial system, rather than trying to maintain the old.
Some additional reading not otherwise noted in the text:
Gibson 1960 "The Aztec Aristocracy in Colonial Mexico"
Horne 1997 Postconquest Coyoacan: Nahua-Spanish Relations in Central Mexico, 1519-1650
A previous related comment here.