"One of the most powerful people in the world is on this airplane" does not seem like something you would want to broadcast, but "Air Force One" is effectively a code-word for that sentence. Why is it used? I would think that presidential flights would have inconspicuous call signs intended to attract as little attention as possible; maybe even fake airline call signs.
From Air Force One:
Columbine II is the first plane to bear the call sign Air Force One. This designation for the US Air Force aircraft carrying the incumbent president was established after an incident in 1953, when Eastern Air Lines 8610, a commercial flight, crossed paths with Air Force 8610, which was carrying President Eisenhower. Initially used informally, the designation became official in 1962.
So the original purpose of the "Air Force One" call sign was to distinguish the special flight from any other flight.
It's difficult to be inconspicuous when you'll also want all other aircraft to keep far away from AF1. Besides that, it's public information what countries the president is visiting. The only special case I can imagine is if the president made one visit to Ukraine. In this case they'll figure out the best way of not broadcasting any security relevant information in advance. Some world leaders have done it.