I'm grateful to the writers from Sydney and Abilene for their comments. They give all of us excellent insight as to how travel forums like Trip Advisor can be so helpful to people like the asker from Chicago.
Travelers are no longer forced to accept "I looked on the embassy website, which says it's impossible," as their final answer. We can now ask those who have been there and actually done what is supposed to be impossible!
Let me add that, having traveled in 44 countries, I've never ever had an immigration officer stop me at random on the street, anywhere in the world, and ask to know my immigration status. It certainly didn't happen to me in Iguaçu National Park.
Why does anyone think the Brazilians would hound people in their national park when it would be a simple matter for the border patrol to turn back visitors at the Neves Bridge crossing, if guides bringing day-trip visitors were thought to be some sort of problem?
The answer is that there will always be some out there who think that just because they haven't done something, it therefore cannot be done.