The league pays for saves because coming in with a one run lead in the 9th is stressful.
Not very more stressful (if at all) then coming into a tied game bases loaded jam with less than two outs in the 7th. But you won't use your best reliever in that situation because it isn't the 9th. You put a lesser guy in, lose the tie, and thus never end up using your best reliever at all as now there's no possibility for a save situation.
And yes I agree that players love the closer designation. As Clip has stated several times, as a pitcher, if you're not a starter, you want to be a closer.
