Q:

Riddle me this, birdman

I wanted to write a riddle in the form of a story, so here we are. Hope you like it.

This is the story of great birds, two.
  Around and around in the sky, they flew.
  One remained small, while the other grew,
  Till they looked like a sparrow, sat on a yew.
The great one went up into the sky,
  Nearly reaching the sun, it was ever so high.
  The heat increased, making the great one cry,
  As it saw its feathers fall off and die.
With a screech of pain, the great one flew to the ground,
  And struck the floor with an almighty sound.
  The great one lay there all heaped in a mound,
  As theother towns called Phoenix in the USA, but the one in Arizona is the best known.)

The first bird is

 an eagle, symbol of the United States.

The business about flying too high, losing feathers, etc.,

 is simply a way to make it a bald eagle, more specific symbol of the United States. There is no particular symbolism to the way in which it loses its feathers, and presumably we should not understand it to have died.

The second bird is

 a phoenix, bursting into flames and then emerging reborn from the ashes. Again, no particular symbolism to the fact that it catches fire from the residual heat of the first bird, nor to the business at the start where the first bird grows while the second doesn't.

