Aztlán was the place they said they came from (probably in today’s U.S. territory). The “Aztecs” called themselves “Mexica”. That’s also their name in Spanish, and I have a faded memory that it is not the name in English only because an anthropologist had trouble pronouncing it or something. Whatever.

Mexicas (meh - SHE - kahs) founded Tenochtitlan. After its fall, you are right, Mexico was named ‘Mexico’ from Nahuatl but people pronounced and pronounce it ‘MEH - hee - koh’ because of the Spanish language influence (think, as in Quixote, ‘kee - HO - teh’).

There were Mexican intellectuals pushing for a ‘meh - SHEE - koh’ pronunciation in the 20th century, but they failed miserably.