Sometimes the problem is not with the Communication professionals, it is the sycophants around the leader who feels the communication professionals are not doing enough or everything must be publicized.
Visibility for leaders is important, absolutely. However, it is not about the leader being visible everywhere and being the talk of everyone. There are a lot of things that could give leaders visibility, farting loudly in front of others, stripping naked etc could make them the talk of the town and very memorable.
Giving a speech in public is important but I dare say what is said is even more important.
Leadership visibility should include aligning everyone to the purpose and vision of your office or the role you occupy. It should be about real conversations on where we want to go, how to get there and providing clarity.
Too much of publicity stunts are dangerous for leadership visibility. Remember the story about the boy who cried wolf?
When next the sycophants are talking to you as a leader, ask them questions on substance and sustainability and see how they respond.