While I believe it is well-established that the Mayans (and other Central American civilizations) practiced human sacrifice and other ceremonies at their pyramids, it always seemed to me that the "stairways to heaven" featured on the sides facing the plazas must be symbolic. Each of the steps are simply too high and steep for any human, certainly anyone in ceremonial garb, to use with dignity and safety. One of the few places you are allowed to climb them today, the Nohoch Mul Pyramid at Coba, can be mounted by most people only with the aid of a rope. The platforms at its side, which the guide said were used for day-to-day visits by worshippers, have normal, modern scales steps, so the Mayans certainly knew what a practical step height was. Moreover, the side of Nohoch Mul features what look suspiciously like the remains of a zigzag ramp which priests could use to reach the summit without risking a broken neck.
Do Spanish or any other sources attest that the stairs were explicitly used?
Re: The Mexica/Aztecs
From the Florentine Codex, a sixteenth century text assembled by a Spanish Benedectine friar and proto-ethnographer towards the goal of establishing that the peoples of America were capable of becoming full humans (Christians) from a childlike pre-Christian state they currently inhabited, we have some references that spring to mind.
Specifically, in Book 2/The Ceremonies, in samples like on page 3 where it is describing slave/captive sacrifice in the ritual of Tlacaxipeualiztli:
...When the masters of the captives took their slaves to the temple where they were to slay them, they took them by the hair. And when they took them up the steps of the pyramid, some of the captives swooned, and their masters pulled them up by the hair to the sacrificial stones where they were to die.
[extended description of the sacrifice process redacted for brevity]
After having torn their hearts from them and poured the blood into a gourd vessel, which the master of the slain man received, they started the body rolling down the pyramid steps. It came to rest upon a small square below.
Now, at first glance, this does appear to suggest the one set of steps in the last part is the same as the one in the first part. However, when one considers that this was put together by a process of consulting groups of people who had witnessed these things/heard of them, and assembling a composite story of how rituals were practiced in their full glory versus the truncated versions of rituals that were causing periodic heresy panics at the time of the Codex's creation, it's easy to see how a strict forensic accounting of which steps were used for what could slip by the wayside.
On pg 26, for example, in the festival of Quecholli, a hunting-oriented festival, it concludes like this:
The hunt ended, they slew captives and slaves on a pyramid which they call Tlamatzinco. They bound them hand and foot, and carried them up the steps of the pyramid (as one carries a deer by the hind and forelegs to the slaughter). They slew with great ceremony etc...
Again ambiguity because of the difference in focus of your question and the chronicler. Here you can easily read this as a practical energy-conserving modern observer looking for the easiest route between two places you need to convey a trussed body to, or you can view it as an opportunity for a slow, ceremonial feat of strength that is on display atop the giant-size steps that are not cut for men but for gods.
It's worth underlining that the Codex was written by someone who was compiling these events as accurately as he could from oral histories of available Mexica groups in his area, using what most modern scholars agree was a good facility with the language as far as understanding it. His focus here would likely be to portray the events accurately, yes, but to highlight instances of penance and other things that could be analogized to the Christian faith, rather than the details you are looking for.
I did take a look at the Maya primary source copies I have on hand, but nothing sprang to mind as straightforwardly as these famous/notorious snapshots that take the trouble to describe the event as it evolves rather than just at its climax, and other Mexica codices with which I am familiar are not really as easily (for better and worse) put to these kinds of purposes. Once you mentioned this, it occurred to me what a relatively rare snippet this is, in my limited experience as a historian but not an expert in the ancient record of Latin American peoples per se, so hopefully others with more specific expertise can weigh in with expertise-sourced opinions of how this would have taken place, if indeed it is possible to elaborate further.