I lived in Mexico for 10 years and traveled a lot everywhere throughout the Country in those 10 years. During a vacation in Maracaibo, Venezuela (where, as many Italians, I had relatives living who had amazing public health system) I had to go to ER and was impressed by their efficiency and system, for free. Cuba, as fucked up and "wrong" as it has been and is, has always had an amazing health system. I also experienced public health system in Argentina and Ecuador.
Believe me, your statement "Mexico, and Latin America in general, have very poor public health care systems" is simply NOT true.