Almost all of that is wrong.
The tune to "
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" was originally for a song called, "Good Morning to You", which was written by Patty and Mildred J. Hill in 1893. By around 1911 (possibly as early as 1901) someone added new lyrics to it, making it into the "Happy Birthday to You" song we know today. However, whoever created that version did not include their name in the credits for the song or any copyright on the lyrics. The Summy Company registered copyright on it in 1935, crediting authors Preston Ware Orem and Mrs. R. R. Forman, based on their particular piano arrangement of the song and added lyrics.
The only reason why they were able to register that copyright was because they claimed that its authors worked for them, thus it was their property. So, no, someone didn't just find it uncopyrighted and slap their copyright on it. They had to demonstrate ownership first.
A division of Warner Music Group, Warner/Chappell Music, bought Summy Company in 1988. However, in 2013 they were sued for falsely claiming copyright of the song and lyrics. This was upheld in 2015, where the judge stated that they only owned the right to that particular piano arrangement, but not the song or lyrics. Thus they lost ownership of the copyright, due to their false claim, and had to pay a settlement of $14 million to those they had previously charged for the song (a mere drop in the bucket compared to what they earned off it).
Furthermore, since they no longer owned the copyright, the original authors were long dead, and no copyright had been filed (as was the law at the time the song was written), "Happy Birthday to You" was now in the public domain in the US. By 2017 the song has become public domain worldwide.
TL;DR: Your own example proves you wrong in that it was ruled as a false claim of copyright in 2015 . Furthermore, "Happy Birthday to You" is
not under copyright anymore.
Copyright is not an exclusively American law. It's accepted by most countries which have agreed to the
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. (Though, admittedly, the application of copyright law does vary somewhat from country to country.)