I disagree about that specific point. It’s one of the meanings of “America”, and is used that way throughout the world. From https://www.merriam-webster.com/dictionary/America:
1 either continent (North America or South America) of the western hemisphere
2 or the Americas the lands of the western hemisphere including North, Central, and South America and the West Indies
3 UNITED STATES OF AMERICA
Similarly in Wiktionary: https://en.wiktionary.org/wiki/America#Proper_noun
“In English, the unqualified term "America" often refers to the United States of America as a synecdoche, with "American" typically referring to people and things from that country. […]”
Regarding the gulf naming, “Gulf of North America” would seem somewhat appropriate, if a mouthful.
"The United States" isn't the name of our country either, does that bother you? Does "US" or "The States?" Those aren't the name of the country either. It isn't in any way exclusionary, it's just a hell of lot shorter and easier to say.
People who call themselves Londoners aren't being exclusionary when they're in the UK anymore than are the Londoners who use that to describe themselves in Canada. They, like Americans in the US, are simply describing what they are, not limiting what other people aren't or can't be.
1: Cf. The Long Now Foundation.
I don't think Google Maps classifying three of the largest countries in the world by area as "sensitive", whatever that is supposed to mean, helps the situation.
Frank Lloyd Wright tried to popularize the name "Usonia" for the United States of North (Independent) America:
You're conflating America with the Americas.
The Americas is the collective name of the continents.
America is just shorthand for the United States of America.
Just like Mexico is shorthand for its official name, the United Mexican States.
And there's no confusion because there's basically never any situation where somebody wants to refer themselves as an inhabitant of the Americas, the same way nobody calls themselves Eurasian. It's too large an area to be useful. It's two continents (in most cultures). Geographically at the largest scale, people tend to refer to themselves as North Americans, South Americans, Latin Americans, and so forth.
+1. To expand on this point, this is different in English vs. Spanish, which is where a lot of confusion comes from.
In English, America is always the United States, and the Americas is always the continents. American always means United States-ian. There is no word for "someone from anywhere in the Americas".
https://en.wikipedia.org/wiki/America https://en.wikipedia.org/wiki/The%20Americas
In Spanish, traditionally, America is the continents, and Las Americas is nothing. Americano traditionally means "someone from anywhere in the Americas".
However, because of the influence of English, people sometimes use Las Americas to refer to the continents and Americano to refer to United States-ians. So it's a little more complicated in Spanish.
https://es.wikipedia.org/wiki/Am%C3%A9rica https://es.wikipedia.org/wiki/Las_Am%C3%A9ricas https://es.wikipedia.org/wiki/Americano_(palabra)
Who proposed the name change from "tissue paper" to "kleenex"?
Gradually, when they didn't enforce their trademark via lawsuits / etc. There's no line in the sand at which point you lose a trademark and it becomes generic. At some point you file suit and fail, or don't even try, and "it is known".
We could be called America if we want but someone would have to propose a name change and it still wouldn't completely co-op the use of the word America to stop having it refer to this entire hemisphere. We all understand that Columbia is a country but Columbia is also a term to refer to this hemisphere which is why we have a place in the United States called Washington district of Columbia. No one confuses that as being a part of the nation of Columbia.
So in the terms of it being the Gulf of the Americas, It absolutely is it's not completely encompassed by one country. We as a nation need to stop calling ourselves Americans as an exclusionary term simply referring to United States citizens. All of us in this hemisphere are Americans but have a different nationalities.