The wings push outward on the air, and via Newton's third law, the air pushes inward on the wings. This produces the centripetal force to turn the eagle's path. So your first friend is wrong, for the satellite does accelerate. Agree with your second friend, for the acceleration of the satellite is toward the center of its circular path, the result of gravitation. Work is done on an object when a force or a component of a force acts along the object's path.
Centripetal force acts perpendicular to the path when an object moves in a circle.
It therefore does no work on the object. Evidence to support this is the regularity of the moon's motion about the earth in its very nearly circular orbit. It moves without appreciable change in its KE or PE because the force of earth gravity is perpendicular to its direction of motion and does no work on it. Newton's first and third laws provide a straight forward explanation. The push by the door provides the centripetal force that keeps you moving in a curved path.