Why doesn't the US have a flag carrier airline?
The definition of a "Flag Carrier", although somewhat vague and frequently disputed, is generally accepted to mean: an air carrier that is owned by, or subsidized by, the government of the country of registration, especially for the purposes of flying international routes.

Wikipedia:


  A flag carrier is a transportation company, such as an airline or
  shipping company, that, being locally registered in a given state,
  enjoys preferential rights or privileges accorded by the government
  for international operations. The term also refers to any carrier
  that is or was owned by a government, even long after their
  privatization when preferential rights or privileges continue.


Business Dictionary:


  Airline ... registered under the laws of a country whose government gives it partial or total monopoly over international routes.




The US has no carriers in this category. All subsidizing of major domestic and international routes ended with the Airline Deregulation Act of 1978. (Note: there are still some small airlines and routes that are subsidized to serve very small communities... If you want to read more about this program, it's called Essential Air Service.)