Perhaps the simples answer is that there were more than just the Tlaxcalans fighting against the Mexica. People from Huexotzinco, Chalco, and even a few from Acolhua, also stepped in to help. To understand how this happened we need to know a little bit about Aztec politics as well as a few key events of the Conquest.

The Mexica Empire is famous (or perhaps infamous) for their sweeping conquest, and while there is some truth to that, diplomacy also played a major role. The nobles of Aztec city-states were divided into groups known as Teccalli/Tecaltin plural (Noble House) which competed with each other for political power in their own state and over their neighbours. One way of augmenting their power was to ally with Teccaltin from more powerful factions. In Central Mexico, this meant allying with the Mexica. Some city-states joined the Empire willingly because being friendly with the Mexica not only protected them from attack, it also allowed them access to exotic goods flowing in from the Mexica's tributary and trade networks. Cholula is an excellent example of this. This system was cheap and effective, and required little oversight from the Mexica. However, it relied heavily on their ability to form, often personal, connections between a huge number of squabbling political factions.

Enter the Spanish. When the Tlaxcalans and Mexica were courting Cortes with diplomatic offers, this was not necessarily a reflection of Spanish military power, so much as it was a continuation of the political influence game that Mesoamerican nations had always played with each other. The goal was probably not to enlist Spanish fighters directly, but to secure an alliance with their employer, the King of Spain, which would guarantee the flow of prestige goods and other material that would bolster each factions claims to domination and legitimacy in Central Mexico. The real threat the Spanish posed was not in their weapons, it was in their lack of investment in this political system. Consequently, they didn't have to play by the rules. This brings me to one of the most important events in the Spanish-Mexica War, the Massacre at the Templo Mayor. This event, occurring in May of 1520, led to the deaths of thousands of Mexica elites, the imprisonment of many others, and the real start of the conflict. The siege and street fighting that ensued within Tenochtitlan forced the Spanish to vacate the city. It is highly likely that they executed many of their noble prisoners as they retreated. There is a good chance that Motecuhzoma was among them. Some of these nobles may not have even been Mexica, but representatives from their subject nations. This was not a new tactic for the Spanish by the way. They had developed this befriend-betray strategy during their conquest of the Canary Islands and practiced it in the Caribbean. There, it had undermined the cohesion of indigenous polities, and it had a similar effect in Central Mexico. The Tlaxcalans could never have done this. They understood the concept of course, and may have directed the similar Cholula massacre. However, the Mexica were wary of them, and would have never allowed them enough access under arms to actually carry it out in Tenochtitlan. The Spanish had no such negative history with the Mexica, and so were able to get much closer.

This was compounded by the 1520 smallpox epidemic, which also killed a vast number of people, 1/4, maybe 1/3rd of the population died. But the real cost was its effect on the Empire as a whole. It destabilised it on pretty much every level, throwing the political networks that Mesoamerican nations relied on into utter chaos. One effect was that the anti-Mexica factions within several nations, such as Acolhua and Chalco, were able to gain power and supported the Spanish/Tlaxcalans. Others were left paralysed, unable to send aid to the Mexica. Others still were basically threatened into supporting the Spanish, only to abandon them when the first opportunity arose. Although the reactions were diverse, the essential cost of the epidemic/assassination combination was to remove the Mexica's advantages in manpower and resources, and transfer them to the Tlaxcalans and their Spanish allies.

Sources:

Berdan, Frances F. and Anawalt, Patricia Rieff: The Codex Mendoza Volume II: Description of the Codex, (University of California Press, Berkeley, 1992) The Codex Mendoza Volume III: A Facsimile Reproduction of Codex Mendoza, (University of California Press, Berkeley, 1992)

Berdan, Frances F., Blanton, Richard E., Boone, Elizabeth Hill, Hodge, Mary G., Smith, Michael E., and Umberger, Emily: Aztec Imperial Strategies, (Dumbarton Oaks, Washington D.C., 1996)

Brooks, Frances J.: ‘Moctezuma Xocoyotl, Hernán Cortés, and Bernal Díaz del Castillo: The Construction of an Arrest’, The Hispanic American Historical Review, 75/2 (1995)

Gutiérrez, Gerardo: ‘Aztec Battlefields of Eastern Guerrero: An Archeological and Ethnohistorical Analysis of the Operational Theater of the Tlapanec War,’ in Scherer, Andrew K, and Verano, John W. (eds), Embattled Bodies, Embattled Places: War in Pre-Columbian Mesoamerica and the Andes, (Dumbarton Oaks, Washington D.C., 2014)

Hassig, Ross: Aztec Warfare: Imperial Expansion and Political Control, (Norman: University of Oklahoma Press, 1998)

Hicks, Frederick: ‘Tetzcoco in the Early 16th Century: The State, the City, and the “Calpolli”, American Ethnologist, 9/2 (1982) ‘Land and Succession in the Indigenous Noble Houses of Sixteenth-Century Tlaxcala’, Ethnohistory, 56/4, (2009)

Restall, Matthew: Seven Myths of the Spanish Conquest, (New York: Oxford University Press, 2003) When Montezuma met Cortés: The True Story of the Meeting that Changed History, (New York: Ecco: An imprint of HarperCollins Publishers, 2018)

There is a difference in the Diamond approach (the cause for the European supremacy can be reduced to their military tech and demographic side effects of disease) and admitting that military technology was a factor, although not certainly the only one. A reductionist approach to history, one that tries to explain all with a single model, has always been problematic and much of the critique on Diamonds work can be traced to that - it's definitely an oversimplification.

Now horses, cannons and armour were advanced technology in the era, and wouldn't have been used so extensively if they played no part. They were "designed" to counter certain feats of European military culture, but certainly found their uses in the Americas too. The primary function of a cavalry charge in medieval warfare is to break the enemy's morale and rout them - few people can hold their ground when armoured soldiers are rushing towards them on horseback. This was very effective against the natives who didn't catch up with the scheme so fast. Cannons busted fortified positions as they did back in Europe. One very key thing was the military culture in general, simplifying a bit you can say the European waged war to kill while the Aztec were known for their ritual war to take prisoners for sacrifice. And indeed they had both been doing this hundreds of years. Military technology is very much the collective refinement of practices, tactics, strategy and culture as much as the weapons themselves.

The Spanish had earlier military successes of smaller scales, which were a factor of convincing the native leaders (and Montezuma himself) of their usefulness. The diplomatic approach of Cortes was basically to claim Charles V was the emperor of the whole world, and more Spaniards would be coming. My sources don't reveal if the Mexica actually believed them, but it was an easy decision for the local noble to swear loyalty to this powerful but distant monarch in return for a powerful ally against the immediate rival. It can't be also underestimated how much he, and Pizarro in a similar fashion, won by striking directly against the person of ultimate power in the region, capturing and assassinating them, and ensuring the collapse of the power structure.