How did the Spanish manage to successfully establish control over Mexico in the immediate aftermath of the conquest of Tenochtitlan if the Tlaxcalans saw the Spanish as auxiliaries in their own war against the Aztecs and had no reason not to see their newly conquered land as rightfully their own?

Oh boy, this is kind of a long one, there's a lot to unpack in this answer, I'll try to go from the general to the specific, but Im not a historian so it may get a little confusing and I might ramble a ~~bit~~ lot, if there's anything that stands out please let me know, and please let me know if Im misremembering something, its been a few years since I read this section of Sahagun's.

In any case please keep Julius Caesar's famous words on the conquest of Gaul here: Divide and Conquer.

Part 1: Tlaxcaltecans before the conquest.

So, first of all, the concept itself of "Right of Conquest" that was so practiced in Europe and the near East (and other places Im sure) wasnt as prevalent in Mesoamerica (it existed, I guess, but not quite in the same way), war was a very ritualized affar, and even when a nation had the means of conquering and controlling another polity it was often a politically bad move to do so inmediately or directly (there are a number of historical reasons for this in the Valley of Mexico), a good example of this is that while the Mexica (Aztecs) could have conquered the Tlaxcaltecans at any point before the conquest of Mexico and had in fact defeated them militarily several times over the years (and quite easily at that as well), they instead (allegedly) instituted a series of ritualized wars with them, the famous Flower Wars (if they indeed existed, which is another whole can of worms), same thing with the rest of the Triple Alliance/Aztec Empire which wasnt so much an empire but more of a hegemony centrally administered by the Mexicans.

Prestige was another important factor, the Mexica were prestigious warriors, diplomats, builders and direct decendants of the Tollan blood-lines all mayor factors in keeping their controll of their empire, the Tlaxcalans? they were a nation famous for their priests, and being good at intrigue (ie. their actions during the war at Huecotzingo), all very good things when trying to destroy a hegemonic empire, not so much when trying to form one.

The quick answer to your question is simply: They couldnt.

But you're not here for simple answers I assume, so now that we have set the stage lets try to dig deeper into the circumstances of the Tlaxcalans during and following the conquest of Mexico, the so called Republic of Indians.

So first of all, you claimed that "the Tlaxcalans saw the Spanish as auxiliaries in their own against the Aztecs", this is not correct, and to explain why I need to go into the Tlaxcaltecan political system and their leader, Xicotencatl. See the Tlaxcaltecans werent an empire, they didnt have a Tlatoani (king) they had a senate representing all the groups within the Tlaxcaltecan kingdom proper (Mesoamericans were big on Confederacies), of which Xicotencatl was their Cihuacoatl (or Tlatoque I dont have my Florentine Codex on hand, but he was the sort-of-kind-of-the Prime Minister, ok?).

See at the beginning of the Conquest many in the senate believed this was a good oportunity to dethrone the Mexica from hegemony and form a new Triple Alliance to dominate Mesoamerica (as had happened in the past), and this King of Spain, whoever he was, sounded like their ticket to do so, indeed many were exited at the prospect, many, but not Xicotencatl, and this important, because the spaniards werent native to Mesoamerica, their way of doing things wasnt normal to them, in fact Xicotencatl many times was wary of the atrocities the spaniards repeatedly engaged in during the conquest.

The turning point came with Otumba, see the Mexica had driven off the spaniards from Tenochtitlan and routed their army, but they were weakened, their leaders killed during the massacre of the Plaza Mayor, their ally Texcoco was refusing their calls for a unified front and worse, a plague had began on their city, they were weakened and so they sent emissaries to 2 other polities, one, to the powerful Tarascan Empire to the north, and another to the Tlaxcalans themselves, they proposed a unified front against the spaniards, the formation of a new Triple Alliance and the split of Mesoamerica between the new powers, the Tarascans refused (they were ancestral rivals of the Mexica), and of course the Tlaxcaltecans... accepted the proposal (bet you didnt saw that coming).

And this is were Caesar's quote comes into full effect, see, Cortez was many things, dumb wasnt one of them, and he had a very good intelligence network, a minority (or so they claim to Sahagun) of the senators wanted to get rid of Xicotencatl and continue the war, they let Cortez knew that Xicotencatl had accpeted the offer and before he could present it to the full senate, Cortez ambushed and killed Xicotencatl according to the sources he maimed the body so horribly that when he paraded it in Tlaxcala and presented it to the senate, they were so scared of the brutality the spaniards were capable of that they accepted to allow Cortez to lead their forces, a similar thing would happen in Texcoco later on, but for the time being, this is an important note, the spaniards werent auxiliaries from that point on, the Tlaxcaltecans were, and everyone made note of that.