Kayne West is widely regarded to be a loose cannon and speak his mind very loudly. He is the opposite of traditional and does not follow social protocol.Just ask Taylor Swift.Whether he is American or not is a different question. That depends on who you ask.Some people think that being loud-mouthed is American.Others think that is goes against American values to insult others.
Yes, Kanye West is not both traditional and American. He is American, but he is not traditional. Being traditional implies that you are conservative to some extent. Kanye West is not conservative by any means, including his profane song lyrics. He has also posted provocative pictures on social media that would not qualify as traditional.
The good thing about America is that there is no such thing as a traditional American. America is the country of diversity. There are as man ways to be a typical American as there are typical Americans. West is as much of an American as anyone else, in his own way.
Kanye West is not traditional and sometimes he does not act very American. West has had a history of making controversial statements--many times these statements are simply to gain attention for himself. Therefore, it did not make sense to invite him to the presidential inauguration. America will be happier with West at home.