You may be interested in this earlier answer of mine, on Aztec organisation and how Tlaxcala at least at first profited from the Spanish alliance (see also especially the follow ups):

• https://www.reddit.com/r/AskHistorians/comments/88a0qu/comment/dwj6qnl

This one goes more into early resistance to Spanish rule:

• https://www.reddit.com/r/AskHistorians/comments/8q2lfr/comment/e0to84e

Some additions for your question:

À big part of why Tlaxcala didn't become more important at that time was that there was only place for one colonial centre in Mexico: Mexico City/Tenochtitlan. Cortes had already early on recognized the former Tenochtitlan as the successor to the Mexica's capital. The Spanish in many ways built on the Mexica's structures, facilitating their rule.

The many individual Nahua states had strong local patriotism with their own political interests. In addition, various states had sided with the Spanish - including former allies (eg partly the Acolhua), former enemies (like Tlaxcala) and former tributaries of the Mexica.

So after the Mexica, the Tlaxcalans were probably the group that profited most from Spanish overrule. They were essentially seen as the Most Important allies even after the conquest, see the 1st answer above for more on this.

Despite this, Tlaxcala had suffered major crises and depopulation by the late 16th century - as had basically all of central Mexico. Factors included forced native labor and resettlements, the introduction of hierarchical Spanish laws and government, of Christianity and the alphabet. Together with increasing European immigration and esp. the ongoing waves of diseases, this led to a massive upheavals especially in the Valley of Mexico into the 17th c.
So I'd say that Tlaxcala at first profited and did keep it's elevated status throughout colonial times. But! It was just as heavily hit by the ravages of colonial rule and disease as most parts of the Region.

Hope this helps!