The key word in that phrase is “Americans” ... as in “citizens”, not just residents, legal or otherwise.
“Ah, but to the Spanish-speaking set, everyone who lives in North or South America is ‘American’. That’s why our border shouldn’t count.”
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True, but to them I’d say any or all or the following:
Show me one person who answers “Where are you from” with “South America”.
Sing me the South American national anthem.
Introduce me to a soldier in the South American army.
Oh, and if borders within those “American” continents don’t count, then the US government’s power to seize property via eminent domain does not stop at the border of any North or South “American” country. OK, thanks for the reminder, so now we’ll seize your lands, enforce our laws, tax you and your neighbors ... oh, and congratulations, since then you are indeed an “American”.