Is The American President the Leader of the World?

I was watching the Democratic Party's debate the other night on YouTube. In the beginning of the debate, candidate Lincoln Chafee said that Americans will be electing the next "Leader of the world." This notion is ridiculous in today's day and age, especially considering that most respected publications now consider China to be the world's largest economy (Link). Also, countries like Germany and Japan have huge economies and are strong influences in the world today. In addition, the American economy is propped up by $600 billion in new debt every year. Sorry Chafee, it's not 1950 anymore.