Don't go to America is the best alternative.

Only go to good parts. Like Mexico and South America in general.

or, perhaps, they are wryly pushing back against the USA capturing a generic geographic term for themselves

To be fair though, that horse bolted a couple of centuries ago. What other name would you call it by? There's another "united states" on the same continent. The country to the immediate south is formally known as the "United Mexican States".