> Why don't we export some better culture to Europe then?

Because there is way more impact potential of doing so from within the US than from Europe.

Also, because moving to another country with the primary intent to "export culture" is not something (I suspect) most people are interested in (I know I am not). Overall, it feels like a weird proposal. And I expect most of the residents of those countries to treat you with both suspicion and bewilderment, if you told them that this was your reason for moving to their country (and rightfully so, in my opinion).

  Take up the White Man's burden— 
      Send forth the best ye breed— 
  Go bind your sons to exile
      To serve your captives' need;
  To wait in heavy harness
      On fluttered folk and wild—
  Your new-caught sullen peoples,
      Half devil and half child.

You couldn’t epitomize the problem more if you tried.