The rest of the world calls it "football" (or kickball, like the Italian name "Calcio", and maybe the Chinese name?)
To be precise, the game being played in 206 BC China was called kickball. If using literal translation Chinese also call soccer "football".
That being said, there is a different Chinese term for those oval shaped balls sports like rugby/American/Canadian/Australian football so there are zero confusion between different sports in Chinese, I would say Chinese got a pass on this matter.
This term doesn't exist in English, so there are no way to differentiate those sports unless full name or "soccer" is used.
I find it weird to popularize the American name
Not just American, but also Canada/Australia. Due to Canadian/Australian football also exists there. Since soccer is a term used to avoid confusion, you would see those English speaking countries with different kind of football would use the term soccer to differentiate them.
Edit: I think South Africa also use the term soccer.
Because North America + Australia has very large number of native English speakers combined, for me it's easier to default "soccer" whenever I am using English so extra communication isn't needed. For other countries that only plays soccer, they probably don't need to deal with the confusion in their own language to begin with.
Also I am pretty sure most English speaker knows what soccer is. So even if they are not North American/Australian, the chance of causing confusion with the term soccer is still lower than football.
Personally, it's less about "popularize the American name" but more about communication efficiency.