Ryan Franklin pitched in two games last year with a leverage index over 4.0, according to Baseball-Reference. It's kind of a misleading measurement, considering a pitcher has a role in his own leverage, but that's how it worked out.
On May 7 he came into a game in Pittsburgh with the Cardinals up 3-2 and the Pirates at bat. There aren't many eighth innings messier than the one he came in to finish; Jason Motte came in to start the eighth, and immediately allowed two singles on six pitches, leaving Andy LaRoche on third base and Andrew McCutchen on first for Trever Miller. Trever Miller came in, got the old 1-5-2 double play, retiring McCutchen at third and LaRoche at home and leaving Garrett Jones at second. Then Miller hit the next two batters, then Franklin came in with the bases loaded and immediately threw a wild pitch to tie the score at three. The Cardinals picked up another lead and Franklin picked up the one-run non-save in the ninth while leaving a baserunner on second. Average leverage: 4.57.
On May 14 Franklin came into a textbook two-run save against the Cincinnati Reds and allowed a run before getting a game-ending double play out of Ramon Hernandez with two on and one out. Average leverage: 4.29.
Most of the games a closer sees just aren't like that. A three-run save is barely leveraged at all, if you do it right; even a one-run save isn't quite so bad, which makes sense considering we expect even replacement-level starters to keep a run off the board in every other inning.
Unfortunately for Franklin, Cardinals fans, and facial hair PR flacks, Franklin has been thrown into close games from the start this year, and he's been awful. Every save opportunity he's gotten has involved a one run lead, and so far through his own initiative and the game situation he's made three appearances with an average leverage over 4.0. The Cardinals have lost all three of them.
Of course, Franklin allowed a home run in his one garbage-time appearance, too.
It's easy to realize just how arbitrary relief pitcher decisions are when you think about how relatively calm we'd be if the Cardinals' run distribution had been a little less closer-unfriendly. A three-run save here and there makes a brutal stretch of closer nightmares into just a regular bad run.
That said, relief pitcher decisions are arbitrary, at some point, by necessity. Franklin himself became a closer, twice, through the closer-by-committee system; he won the job both times because somebody else, a veteran like Jason Isringhausen or a rookie like Jason Motte, couldn't quite do it.
As a result I'm not sure how much credit we should give inertia in decisions like this, where we have so little data in either direction. In general small-sample sizes are a terrible thing from which to make personnel decisions, but sample sizes for closers are always small; add his bad start to 2011 to his solid 2010 season and it's the difference between a 3.46 ERA (an ERA+ of 113) and a 4.00 flat, which probably would have lost him his job in the offseason.
A change also makes sense just because the closer role really doesn't. If you demote a starter he has to pitch in a completely different role or the minor leagues; demote a shortstop and he gets to play three times a week. Demote a closer and he might pitch more frequently; it's even possible he might pitch in higher leverage situations. When John Mozeliak says "the closer's role is about confidence" he's right, because if it weren't for the need for confidence at all costs—primarily for the fans, the manager, the manager's bosses—the role wouldn't exist, and managers could just bring in whoever seemed best suited to whichever situation.
Closing is a weird job; it's only tangentially related to winning games and the rules of baseball, and the usual analysis is, as a result, hard to deploy here. If Franklin doesn't inspire confidence, he's not successfully doing this weird job we've carved out for him, and the Cardinals should try somebody else until he does. It's not like pushing Albert Pujols down the lineup following a 2-20 slump; it's like pushing Albert Pujols out of the Hyundai Long Drive Inning slot following a 2-20 slump a year after he hit 15 home runs in 180 at-bats in the role, in some weird Demolition Man universe where it exists.
I'm just saying: Last time the Cardinals tried the whole closer-by-committee thing it produced a guy who saved 82 games and made an All-Star team. What could go wrong?