I was hoping this would draw more attention...

I'm not sure this is all that useful but here is an n-gram plot that shows the usage of "United States of America" in a corpus of books. It's interesting to see a sharp increase right around 1776.

N-gram frequency of "United States of America"

There is a similar pattern for "America."

Many frequencies may be "flukes" because of the phrase being contained within copyrights and such.

Hope it's somewhat useful!