There’s an old line about Heinrich Schliemann, the man who discovered Troy, speaking better Greek than any of the men he hired to help with the dig.
Most illegal Latino immigrants come from poor Mestizo or Uto-Aztecan backgrounds; their families’ grasp of proper Spanish is probably pretty marginal to start.
The Uto-Aztecans should go to Utah--see if they could communicate with the Utes by trying out some Nahuatl phrases on them.
You are absolutely correct. Add to that the incredible amount of idioms and regional/slang expressions found in Mexican Spanish (similarities in Guatemala and Honduras) and the enduring use of and mixture with Nahuatl varieties...and you have a veritable soup of communicative twists. For this reason, you also find that many speakers of Spanish from countries in South America (as an example) are constantly stereotyping and making fun of that “soup.”