I use the term The Americas all of the time, and yes it means all of North, Central, and South America, but I think it’s not necessarily obvious that The Organization of American States includes all of The Americas, because in America we rarely use the term state to describe Latin American countries.
As an aside, my SIL is Mexican and her first Husbands family was Italian. They always called her La Americana, which means The American. Basically, the same way we would call an Italian European.
Also noteworthy, in Spanish The USA is most often referred to as Los Estados Unidos, The United States, not America.
One more item you might find interesting, many countries consider the Americas to be one continent, unlike the US that teaches that it is two. Americans are taught there are 7 continents, many countries use the 6 continent model, some use the 5 continent model, like the Olympics (the 5 rings in the logo).