Those search firms that charge both agencies and their clients for their help are doomed to lose their objectivity. And, if it were up to me, they would lose all their business too. Their way of doing business, if not illegal, is damn close to it. If you are doing a search right, you will uncover a whole bunch of new agencies each time you refresh your database. That is the way of things.
Agencies come and go.
You have to stay on top of that. In the vast majority of cases, incumbents do not belong in a search. They are doomed from the start. One of the primary tasks of the search consultant is to keep the playing field level. Each agency should have an equal shot.