There's two big misconceptions that are commonly held about post-Conquest Mexico: that Hispanicization progressed rapidly and the native population faced discrimination equally. I covered in a previous post how life in Mesoamerica basically continued on with business as usual for almost a generation after the fall of Tenochtitlan. This was doubly true for the Tlaxcalans since they were not only granted great leniency in "New Spain" in maintaining independent governance, but also the end of what was basically an Aztec embargo on trade.

The point made again and again is that, when the Spanish arrived, the Tlaxcalans had neither salt nor cotton, those two items standing in for a whole range of trade for not only luxury, but staple goods that the Aztec encirclement had cut off. Hassig, in Aztec Warfare makes the point that the xochiyaoyotl (flower wars) which had typified conflicts between the Aztecs and Tlaxcalans had, in the years leading up to the Spanish arrival, been becoming more intense, more bloody, more vicious, and more territorial. The whole idea of the Tlaxcalans manipulating the Spanish into the Cholula Massacre was that it had recently flipped from being sympathetic to the Tlaxcalans to shifting allegiances to the Aztecs. Since Cholula sat on the doorstep of Tlaxcala, this was a major blow to their state security.

Thus, after the Tlaxcalans and Spanish spent a few rounds beating the crap out of each other, they eyed each other up and entered into a mutually beneficial alliance. To suggest that the Tlaxcalans manipulated the Spanish to their own gain is, well, to understand the political landscape of Postclassic central Mexico. To suggest that the Tlaxcalan goal was the conquest of the Aztec states is to misunderstand the restricted role they were in. Fighting the Aztecs was a much a matter of survival for the Tlaxcalans as it was for the Spanish, more so even, since the actual existence of the Tlaxcalan nation was on the line.

There are numerous instances in the Spanish texts and even in the Historia de Tlaxcala, written in the 16th Century by a Tlaxcalan-Spanish chronicler, Diego Camargo, supporting the idea that this was a mutual alliance that was highly valued on both sides. For instance, after the deal was struck, Cortés said that they should destroy their "idols," to which the Tlaxcalans basically said, "no thanks."

Gomara's biography of Cortés actually deals directly with this in a chapter titled "The Tlaxcalans Defend Their Idols" wherein the response of the Spanish to the Tlaxcalans rejecting Christianity is thus:

Cortés answered them, promising that he would send someone to teach and indoctrinate them, when they would see the improvement and the very great profit and pleasure they would have by following his friendly advice; but that at the moment he could not do so because of his haste to get to Mexico.

Yeah, his "haste to get to Mexico."

Cortés, who spent weeks on the Gulf Coast casting down idols and messing with local politics before trekking inland through unknown and hostile lands, is suddenly in a huge rush to get to Tenochtitlan. Just in cast the sarcasm isn't coming through, the modern interpretation of this is that Cortés looked around, saw he had zero chance of forcing conversions without alienating the only friends he had, and opted to table the whole Jesus-thing for a later date.

Other notable events include the famous joint "Tlaxcala! Castile!" cheer and the fact that a powerful noble, the son of one of the Tlaxcalan rulers (like the Aztecs, Tlaxcala was not a singular state but a confederation of altepetli tied together by custom and marriage), was put to death for his agitation against the Spanish following their expulsion from the Valley of Mexico after La Noche Triste. Here is Bernal Díaz del Castillo's account of a speech given by Maxixca condemning the noble in question, Xicotencatl the Younger:

I ask you, do you yourselves think, or have you ever heard others say that such riches or so much prosperity was ever known for the last hundred years in the land of Tlascalla as since the time these teules [the Spanish] have appeared among us? Were we ever so much respected by all our neighbours? It is only since their arrival we possess abundance of gold and cotton stuffs; it is since that time only we eat salt again, of which we had been deprived for such a length of time. Wherever our troops have shown themselves with these teules, they have been treated with the utmost respect; and if many of our countrymen have lately perished in Mexico, they certainly fared no worse than the teules themselves. All of you must likewise bear in mind the ancient tradition handed down to us by our forefathers, that, at some period or other, a people would come from where the sun rises, to whom the dominion of these countries was destined. How dare Xicotencatl, taking all this into consideration, contemplate this horrible treachery, from which nothing can flow but war and our destruction? Is this not a crime which ought not to be pardoned? Is it not exactly in accordance with the evil designs with which this man's head always runs full? Now that misfortune has led these teules to us for protection, and that we may assist them with our troops to renew the war with Mexico, are we to act treacherously to these our friends? (Lockhart trans.)

You can see the lines about having access to salt and other goods again, but also the idea that "we are in this together," which really does permeate a lot of how the Tlaxcalans and Spanish come off during this period. At the end of the war with the Mexica (in which the Tlaxcalans are noted as engaging in a bit of looting), this paid off in being relatively unbothered by the new Spanish regime. It was a return to normalcy. As Lockhart notes in Nahuas After the Conquest, the fact that Tlaxcala was saved from the ravages of war meant that it was also the best organized state in the region, and thus provided a great deal of officials, like traveling judge-governors, throughout central Mexico. Groups of Tlaxcalans also moved north with Spanish/Aztec forces in the pacification of the Gran Chichimeca to establish towns in what are now San Luis Potosí, Coahuila, and Zacatecas. They essentially had an independent state which, while denied tributary domination over the nations in the Valley of Mexico, nonetheless benefited from no longer being under siege and branching outward.

So why didn't the Tlaxcalans end up establishing dominance over the Valley of Mexico. The easy answer is that the Spanish moved very quickly to recognize Mexica nobility and re-establish them in their roles. Descendants of Motecuhzoma II continued to rule Tenochtitlan for decades after the "conquest," and there is in fact a Spanish ducal title held by his direct descendants. This lazy ethnocentric answer fails to take into account that, with the surrender of Cuauhtemoc, the Spanish/Tlaxcalans no longer had any cause to press the war further. Despite beating the Mexica back, the idea that Spanish/Tlaxcalan/Etc. forces would completely remove the Mexica from power would have meant a kind of total war that was really outside the discourse. Within the framework of Mesoamerica, the Spanish had made the Mexica a tributary state and the Tlaxcalans had removed the Mexica as a threat. It was win/win.

Let's take a moment, however, and consider another, more speculative reason why the Tlaxcalans did not press for more substantial spoils of war. Well, one reason is that the lands that served as the buffer between Tlaxcala and the Aztecs, Huexotzinco and Cholula, both allied with the Tlaxcalan-Spanish (albeit after the aforementioned massacre at the the latter altepetl). Then there's the fact that the eastern side of the Valley of Mexico and the 2nd most important state in the Aztec Triple Alliance, the Acolhua, flipped to the Spanish early on. Thus, the Tlaxcalans could not exactly demand that land. Other groups in the region, like the Xochimilca, also turned against the Mexica, if not exactly jumping on the Tlaxcalan-Spanish bandwagon. Thus those groups were also not considered to be in conflict with the Tlaxcalan-Spanish forces, which again put them off-limits for punitive measures according to the custom of war in both cultures.

A more speculative view is that, following the 1520 smallpox epidemic, the Tlaxcalan leadership was devastated and unable to effectively press their claims. Accounts generally credit Xicotencatl the Elder and Maxixca as the most important leaders. Xicotencatl most likely died either at the tail end of the conquest or shortly afterwards, while Maxixca is said to have died in the smallpox outbreak leaving some sons which are generally unnamed in the sources as successors, or at least what the Spanish recognized as successors. There's a passage in Torquemada (start at the top of pg. 63) which gives clues to the kind of chaos of succession that followed the death of Maxixca. So one interpretation would be that there was no effective leader to press Tlaxcalans claims post-Conquest, and that by the time there was, the epidemics of 1540s and 1570s would devastate the native population, setting the stage for Spanish control.

If I may add a tangent to this question - what rewards did the Tlaxcalans receive from the Spanish after the Aztecs were defeated?