I completely agree that adding these polarizing terms is not helpful in persuading anybody, however I think it's an extreme simplification to say the American Left and Right are leanings of centrism from the view outside the US.

This is kind of an outside perspective - America as seen through the news - but I would say that a lot of the stuff I see espoused by the Left and Right in the US is much more extreme than in Germany for example. Germany has laws that allow for positive discrimination, but I can't think of any such extreme measures as in the US here, where chances are diminished for one race by a factor of 10. Even having race on the application form would be unthinkable here.

On the other hand we have many more social programs that would be unthinkable in the US. We have 4+ (usually 5-6) weeks vacation per year, etc.

The American Right on the other hand is also in many ways more extreme than in Germany, nobody would consider Republicans as right-leaning centrists here, e.g. building a wall would be a bit taboo, although maybe more due to historical reasons. Denying people health-care, because they're poor would also probably not count as centrist here.