Agree. I would add (if I understand your point and I think I do) that putting a resume out is somewhat deprecating to a famous person. If Rob doesn't land a job, as happens with unsold real estate, potential employers (aware of the public search) will wonder why he hasn't been snapped up by someone else quickly. By going public you run this risk. But doing the search in private (or as they say "quietly shopping") you avoid some (but not all) of the risk. If nothing materializes you can go public later.
"you're competing with other resumes"
Very true. One of the best ways to not be compared is to approach things in a way that there is no competition. Approach employers when they are not advertising. I did this once and got a job that in no way I was qualified for resume wise. Once a job is advertised or with a headhunter you're being compared to someone else. (Same with dating most of us would rather not have to compete head to head with 20 other people and hope to be chosen the best of the lot.)