So it turns out that the records are pretty spotty since the Aztecs kept no written text of their myths and they simultaneously held several oral forms of the same stories (kinda similar to how early Israelites likely were). But there is one solid written source by the grandson of Montezuma II (made famous as the Aztec ruler during the Spanish conquest) from 1609.
While the text seems to make no big deal of Huitzilopochtli talking to his people he does do so, though more often he speaks through intermediaries the 'idol bearers.'
"Here it is told, it is recounted
How the ancients who were called, who were named,
Teochichimeca, Azteca, Mexitin, Chicomoztoca, came, arrived,
When they came to seek,
When they came to gain possession of their land here,
In the great city of Mexico Tenochtitlan. . . .
In the middle of the water where the cactus stands,
Where the eagle raises itself up,
Where the eagle screeches,
Where the eagle spreads his wings,
Where the eagle feeds,
Where the serpent is torn apart,
Where the fish fly,
Where the blue waters and the yellow waters join,
Where the water blazes up,
Where feathers came to be known,
Among the rushes, among the reeds where the battle is joined,
Where the peoples from the four directions are awaited,
There they arrived, there they settled…
They called themselves Teochichimeca, Azteca, Mexitin.
They brought along the image of their god,
The idol that they worshipped.
The Aztecs heard him speak and they answered him;
They did not see how it was he spoke to them…"
As you can note, it specifically says that the Aztecs heard him speak, though they did not see how - kinda like the the revelation in the Torah, hmm?
The last chapter goes like this:
"The Culhuacan pursued them, they pursued the Mexicans,
They drove them into the water….
The Culhuacans thought that they had perished in the water,
But they crossed the water on their shields,
They crossed on their arrows and shields.
They bound together the arrows, called Tlacochtli,
And those called Tlatzontectli,
And, sitting upon them, they crossed the water….
And sitting upon the shields they crossed the water
When the Culhuacans pursued them.
And they came into the rushes, into the reeds at
There they dried their battle gear which had become wet,
Their insignias, their shields—all their gear.
And their women and children began to weep.
They said, "Where shall we go? Let us remain here in the reeds…."
And then the old Mexicans, Quauhtlequtzqui, or Quauhcoatl,
And also the one called Axolohua went off,
They went into the rushes, into the reeds
At the place that is now called Toltzalan, Acatzalan;
The two of them went to look for the place they were to settle.
And when they came upon it,
They saw the many wondrous things there in the reeds.
This was the reason Huitzilopochtli had given his orders to the idol-
bearers, his fathers,
Quauhtlequetzqui, or Quauhcoatl, and Axolohua, the priest.
For he had sent them off,
He had told them all that there was in the rushes, in the reeds,
And that there he, Huitzilopochtli, was to stand,
That there he was to keep guard.
He told them with his own lips,
Thus he sent off the Mexicans.
And then they saw the white bald cypresses, the white willows,
And the white reeds and the white rushes;
And also the white frogs, the white fish, and the white snakes
That lived there in the water.
And they saw the springs that joined;
The first spring faced east and was called Tleatl and Atlatlayan,
The second spring faced north and was called Matlalatl and also
And when they saw this the old men wept.
They said, "Perhaps it is to be here.
We have seen what the priest, Huitzilopochtli, described to us
When he sent us off.
He said, `In the rushes, in the reeds, you shall see many things.'
And now we have seen them, we have beheld them!
It has come true, his words when he sent us off have come true!"
Then they said,
"O Mexicans, let us go, for we have beheld them.
Let us await the word of the priest;
He knows how it shall be done."
Then they came, they sojourned in Temazcaltitlan.
And during the night he saw him,
Huitzilopochtli appeared to the idol-bearer, called
Quauhtlequetzqui, or Quauhcoatl.
He said to him, "O Quauhcoatl, you have seen all there is in among
In among the rushes,
You have beheld it.
But hear this:
There is something you still have not seen.
Go, go and look at the cactus,
And on it, standing on it, you shall see an eagle.
It is eating, it is warming itself in the sun,
And your heart will rejoice,
For it is the heart of Copil that you cast away
Where you halted in Tlalcocomoco.
There it fell, where you looked, at the edge of the spring,
Among the rushes, among the reeds.
And from Copil's heart sprouted what is now called Tenochtli.
There we shall be, we shall keep guard,
We shall await, we shall meet the diverse peoples in battle.
With our bellies, with our heads,
With our arrows, with our shields,
We shall confront all who surround us
And we shall vanquish them all,
We shall make them captives,
And thus our city shall be established.
Where the Eagle Screeches
Where he spreads his wings,
Where the Eagle feeds,
Where the fish fly,
And where the Serpent is torn apart.
And many things shall come to pass."
Then Quauhcoatl said to him, "Very well, Oh priest. Your heart has
Let all the old men, your fathers, hear."
Then Quauhcoatl gathered the Mexicans together,
He had them hear the words of Huitzilopochtli;
The Mexicans listened.
And then, once more, they went in among the rushes, in among the
To the edge of the spring.
And when they came out into the reeds,
There at the edge of the spring, was the Tenochtli,
And they saw and Eagle on the Tenochtli, perched on it, standing
It was eating something, it was feeding,
It was pecking at what it was eating.
And when the Eagle saw the Mexicans, he bowed his head low.
(They had only seen the Eagle from afar).
Its nest, its pallet, was of every kind of precious feather—
Of lovely cotinga feathers, roseate spoonbill feathers, quetzal
And they also saw strewn about the heads of sundry birds,
The head of precious birds strung together,
And some bird's feet and bones.
And the god called out to them, he said to them,
"O Mexicans, it shall be here!"
(But the Mexicans did not see who spoke).
It is for this reason they call it Tenochtitlan.
And then the Mexicans wept, they said,
"O happy, O blessed are we!
We have beheld the city that shall be ours!
Let us go, now, let us rest…."
This was in the year 2-House, 1325."
It is very interesting to note the similarities to the Torah here as well. The nation escapes an enemy army over a body of water and then the people just want to call it quits. Then the whole nation upon traveling further sees the amazing vision of a shiny albino land which was prophesied and their god says aloud "O Mexicans, it shall be here!" and the people are overjoyed.
So yes, the source I found before was a rather modern romanticized version of the classic Aztec migration myth, but the original still has enough to undermine the Orthodox apologetic claims that no other people has ever had a national revelation besides the Israelites.