No these a..holes work around that by claiming they are the Aztecs...
On the contrary, the Aztecs and the other pre-Columbian Mexican indigenous populations were never wiped out. In fact, their descendents make up the vast majority of the present-day population of Mexico. That is what gives rise to the entire La Raza/Aztlan mentality. The way they see things, they are the Aztecs.
These individuals do not identify with the Spaniards that landed in Mexico in the 1500's. They identify with the indigenous population of pre-Columbian Mexico and stress, as this link points out, that "85-90% of the Mexican population is of Native blood."
As such, they glorify the Aztecs (although the Spaniards conquered the Aztecs with the help of all the smaller Mexican tribes that were fed up with having their hearts cut out in Aztec religious ceremonies) and villify what they consider "European".
This attitude then gets transferred to anti-Americanism as they view Americans as just another group of "European imperialists".