Because you change the role all around when you do that. Most modern closers start their appearances with the bases empty. Most of them get prepared to start the 9th inning. They would have to adjust and adapt to pitch the 7th. If you're adjusting usage patterns to maximize leverage then your closer will often come into big jams in the last third of the game. Constantly coming in with two on, one out, tying run on first... and "failing" a much larger percentage of the time. Imagine Hunter coming in and pitching like he did yesterday, but, say, after the starter had walked the first two batters in the 7th. Runs would have socred, and Hunter would have failed to protect the lead. And then you're left with your 2nd or 4th or 7th-best pitcher to pitch the 9th, which will still come quite often with your team within a run or two either way.
When do you warm up your best reliever? In the 7th when the first batter walks? Then what happens when the next guy hits into a DP? How do you avoid warming up someone multiple times trying to get him ready for the theoretical max-leverage point?
Is it theoretically possible to have your best pitcher pitch only in high-leverage situations from the 6th-on? Probably. But it's going to be harder for everyone, and the gains will likely be small. If they exist at all. You will certainly have many games where you use your best reliever in the 7th, only to find that a higher-leverage situation comes up later.
People may not like the designated closer strategy, but it's a pretty fair proxy for "strategy that's not impossible to manage while still getting your best reliever the highest LI in the game most of the time."
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