Of course the only reason we use the term "American" is because of the USA! It's a natural term for residents of the USA, and has become ingrained in English by virtue of America being the largest English speaking country for nearly 200 years. Do non-English speakers have a problem with this? In Spanish the term for USA is "Estados Unidos" which isn't problematic at all.
It has nothing to do with language, it's about regional identity. Imagine if the French told Europe that France is now Europe, and what was Europe will now be called the Europes. It wouldn't matter if they say it in French or German.
