Not to mention the United States of Brazil. But that's an argument on behalf of the term "American." And Mexicans don't even think of themselves as "Americans," let alone identify themselves as such, so why are you inventing a fictional identity for them?
You remind me of the reporters who change the words of immigrants (legal and otherwise), and claim they all said they wanted to come to "the United States." I've never heard an immigrant talk of coming to "the United States." It was always "America." There is only one nation on the face of the earth named "America." You can deny that all you want, just as you can insist, if you like, with an equal lack of justification, that you are Julius Caesar.
I'm not sure how many immigrants you speak with from Latin American countries. But having lived in South America, and having personally spoken with thousands of Mexicans, Central- and South-Americans, I can assure you that they DON'T refer to us as "America." We are "los Estados Unidos" to them, and we always will be. It's just the accepted norm in Spanish. And, they feel that everybody in this hemisphere is an American.
(That fine semantic point out of the way, they still don't have the right to cross our borders illegally.)