The major reason that Churchill was the one to give this speech is that he was a renowned world leader who wanted to push the United States to pursue a given foreign policy.
The Iron Curtain speech was not a statement of US policy. That would have been something that the president would have done. Instead, it was Churchill's own opinion as to how the United States should react to the Soviet Union. Churchill was given the chance to give this speech because he had been the Prime Minister of England during World War II. Because of his stature in the world, and because the speech was not a statement of US policy, Churchill was able to give this speech.