A little column A, a little column B. This is a massive area of debate and I'll do my best not to butcher it. Hopefully some wandering Mesoamerican flairs will turn up to enhance the picture.

A few pre-contact codices such as the Codex Borgia survived the conquest and its aftermath, with the Codex Borbonicus arguably the most notable example. It's the only one I can recall that was likely commissioned before or during the conquest, for a native patron, that could have originated from Tenochtitlan. However, even this is disputed, as Spanish annotations have been added and in the instance of the calendar pictures space seems to have been deliberately left for them in the original production. The Codex Borgia is less likely to have suffered from European interference and is assumed to have originated from the Puebla area to the east of Tenochtitlan.

Both of these are probably pre-contact 'writing', but not as Europeans would recognise it. They are pictorial in nature and would mainly have served as divination aids for priests and as a ready depiction of the Aztec pantheon. What literature that exists in the European sense of the word (alphabetic script that dictates pronunciation as well as meaning through spelling) is neccessarily affected by European influence as the Aztecs simply didn't have (or arguably even need) an alphabetic script to do so with.

Therefore, what written sources do exist, to quote Clendinnen, "were drawn afresh, to satisfy Spanish curiosity and to answer Spanish questions." Whilst indigenous sources were undoubtedly involved in the production of the codices Florentine, Mendoza, etc., they are ultimately "hybrid affairs... deeply if unevenly tinctured by Spanish Christian assumptions and expectations." We actually have a decent variety of sources of this nature, if not exactly a profusion of volume. Everything from imperial tribute lists to official histories to accounts of day-to-day life to religious beliefs have been recorded. The question is, as you rightly point out, how much of it is genuine? How much can be trusted?

It seems certain however that in the Florentine Codex at least (the major source for information on Mexica life in the years before the conquest) we are confronted by "Mexica views... accounts of Mexica action as described by Mexica voices", and if Sahagún was restrictive in some areas as a chronicler, in others his sources positively wax lyrical. Superstition and humour penetrate the rich descriptions of daily life and religious ritual, in contrast to the more stuffy official chronicles or tribute lists, or indeed the rather opaque and allegorical Cantares Mexicanos. The subaltern voice is present in a frank and discursive style, and provides glimpses of how the Aztecs might have viewed the world that existed outside of the temple or the battlefield.

Ultimately, the answer to "What do we know of their myths and stories?" is down to the interpretation of the historian in question. These written sources concerning life before the conquest were composed anything from a few years to generations after the event, under massively varying instructions and circumstances. Can we ascertain an accurate idea of Aztec culture, history, society, public and personal lives from the surviving codices? I am confident we can. I am less confident about proclaiming for certain just how much this picture has been influenced by European methodology, psychology and theology. It's just really bloody hard to tell.

Inga Clendinnen, Aztecs: An Interpretation Cambridge, 2014, pp. 387-391

/u/pseudogentry has already covered the major points, so I'm just going to shamelessly repost (with minor edits) a comment of mine from a similar question asked recently.

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I don't have my copy of 1491 handy (the vagaries of lending), nor am I a classicist with an interest in ancient Greek literature. The claim itself is a bit bizarre and I would be interested in the exact passage from Mann, who tends to cautious in making such sweeping claims. I have, however, read a large amount of Nahua literature, of which there is -- to many people -- a surprising amount.

Pre-Conquest writings are extremely rare, given both their initial scarcity and the propensity for the Spanish to toss anything they found in the fire. The Borgia codices are our only legitimate sources, and they are pictorial works, not script. Much is often made of the lack of written language in Mesoamerica, with caveat that the Classic Maya did have a functioning system of writing. While this did not spread to the Aztecs (several hundred centuries and kilometers away), the pictorial system employed by the Aztecs was much more sophisticated than is often given credit. Late Postclassic writing systems are a bit of a distraction though, so let's circle back to the corpus we do have.

The peri-Contact period, along with the following century, gives us a boon of works in Nahuatl (the language) by Nahuas (the overall cultural group), though not all by Mexica (a specific Nahua group). I would assume that Mann is taking a bit of license in lumping literature done by various Nahua groups under the rubric of the Mexica, who were the dominant Nahua group within the Aztec alliance and overall. These early writings generally divide into two main groups:

1. Writings by Spanish clergy

Unsurprisingly, the Spanish friars who arrived in Mesoamerica were keen to convert the populace to Christianity, almost as if it were their job. In order to accomplish their conversion, the Spaniards aimed to learn Nahuatl in order to proselytize in what was the lingua franca of Mesoamerica. These efforts yielded a few key works which are essentially language training books. Friar Molina produced the earliest and most important of these, immediately after the Conquest in the couple of decades when Nahua culture was largely intact. Carochi, a Jesuit friar, would produce another important work a century later.

Molina and Carochi were not the only Spanish clergy working to learn and catalogue Nahuatl, and it is certainly debatable as to whether their works can fit into a notion of "Nahuatl literature." I am fairly certain that scholars of Classical Greece would be more than happy to have a grammar of the language written at the time. Nevertheless, we also have a significant body of Nahuatl literature in the form of the General/Universal History of the Things of New Spain compiled by Friar Sahagún.

Though he was the primary writer and editor, the work (which is basically an encyclopedia of Nahua life) was assembled over the course of years through interviews conducted with ethnic Nahua scribes training at the Colegio do Santa Cruz in Tlatelolco. As such, the work, though not free from Spanish bias, is seen as capturing an authentic Nahua view.

That being said, Sahagún has been accused (by Terranciano, I want to say) of being... encyclopedic, in his diction. Basically, the claim is that Sahagún overwrote in Nahuatl, trying to capture multiple variations of saying the same thing. Nevertheless, we still have voluminous passages in the Historia which can be quite moving, such as this portion on the sacrifice of the ixiptla of Tezcatlipoca:

Thus it was said: when he arrived where the [ixiptla of Tezcatlipoca] used to die, where a small temple called Tlacochcalco stood, he ascended by himself, he went up of his own free will, to where he was to die. As he was taken up a step, as he passed one, there he broke, he shattered his flute, his whistle.

And when he had mounted all the steps, when he had risen to the summit, then the offering priests seized him. They threw him upon his back on the sacrificial stone; then one of them cut open his breast; he took his heart from him; he raised it in dedication to the sun.

For in this manner were all captives slain. But his body they did not roll down; rather, they lowered it. Four men carried it.

And his severed head they strung on the skull rack. Thus he was brought to an end in the adornment in which he died. Thus his life there ended; there they terminated his life when he went to die there at Tlapitzauayan.

And this betokened our life on earth. For he who rejoiced, who possessed riches, who sought, who esteemed our lord’s sweetness, his fragrance -- richness, prosperity -- thus ended in great misery. Indeed it was said: “No one on earth went exhausting happiness, riches, wealth.”

Still, we have to go back to the question of "what is Nahua literature?" And that brings us to the other major sources of writing.

2. Nahuatl writings by Nahuas

Unsurprisingly, once literacy (in the Spanish style) spread throughout the Nahua elite, they wanted to write things down. There are several ethnohistorical texts by Nahuas who wrote dual Nahuatl-Spanish works. These texts are a blend of annals, geneaology, history, and ethnography, often along a particular ethnic slant. The Mexica, Fernando Tezozomoc, for instance, focused on the history of the ruling dynasty of Tenochtitlan, from whom he was descended. Meanwhile Chimalpahin focused on his own Nahua group, the Chalca. Fernando Ixtlilxochitl, an Acolhua, similarly focused on the history of his own family, the ruling dynasty of Texcoco. His work is interesting not only for it's hagiography of Nezahualcoyotl, but Ixtlilxochitl's attempts to square his ancestor's religious beliefs and practices with his own Christian faith, which leads him to claim that Nezahualcoyotl abhorred sacrifices and was a monotheist.

Taken together, however, the ethnohistorical works produced by these authors yield us hundreds of continuous pages of Nahuatl literature. Again, while I am not a classicist, I remain confident that they would very much enjoy have continuous book-length works with clearly identifiable single authors in which the original copies can be identified and research, as opposed to fragments with dubious authorship which have been copied and recopied over the millenia.

All of this is a bit of a tease though, since I have yet to address one particular major work:

3. Cantares Mexicanos

The Cantares are a collection of almost 100 Nahuatl poems. The extant manuscript is dated to the early 17th century, but the poems themselves are from the peri-Contact period. While the poems are biased by their focus on elite men reciting about elite manly things (i.e., they are mostly about war and death), the Cantares are a dense collection of Nahuatl literature. It is a singularly broad and deep collection of literature.

So why isn't this body of literature better known to English speakers?

The first problem is that many of the core works of Aztec history have never been translated into English, or have only been (relatively) recently translated. Fernando Tezozmoc's Cronica Mexicayotl is available in Spanish, but not English. Same with Fernando Ixtlilxochitl's Historia Chichimeca; Spanish language version are available, but nothing in English. Sahagún's Historia was only fully translated into English in the 1970s, in a process that took years. The works of Chimalpahin only made it into English in 1997, with Dibble and Schroeder's Codex Chimalpahin. The Cantares translation and exegesis were published by Bierhorst in 1987. So one solid factor in the Anglophone world's ignorance of these texts is that they literally did not exist in English until recently. Compare and contrast that with English translations of Cortés' letters or Díaz del Castillo's True History, which have had English translations available for more than a century.

The other factor is that, in the Anglosphere and in American/European history in general, the Greek works are seen as foundational texts. Ancient Greece is seen as the foundation of European civilization, by way of the Romans. While that view elides over the Near East/Western Asia orientation of the Greeks, it also have some truth in that the ancient Greeks became a touchpoint many successive groups. As such, their works were translated and copied and passed from hand to hand in a way that preserved them to this day (albeit imperfectly and incompletely).

Honestly, the notion that we have more surviving texts from a group 500 years ago, rather than 2500 years ago, should be completely unsurprising. The bias towards Greek writings as being foundational texts, however, means that they are not available in myriad languages, but are also taught as part of ethnohistory. While American, or British, history may have more immediate connections to the ventures of Cortés et al., the fact that Athenian democracy has such symbolic power and Platonic/Aristotelian philosophy had such profound impact on later European societies means that history and literature are incorporated into those countries' basic curriculum taught.