What's the real shame is that there has to be a set closer at all. Ideally you would use your best relievers in the highest leverage situations, regardless of whose inning it "belongs" to.
I think the real pressure isn't coming from management, honestly, but from the players. As long as the league pays for saves nobody who's got a taste of closing and been successful at it is going to be satisfied in any other role. I half-expect Clip to bolt as soon as he's an FA for that very reason.