If you can't pitch in a tie game in the 9th (or extra innings) as a closer that's a problem. I don't think anyone has ever suggested that's a thing, versus the idea that closers don't know how to pitch in non-leverage, non-stress situations. But it's all anecdotal.
I still prefer the no-closer bullpen, but it is certainly much easier on the manager to not have to think about it, and obviously many of the pitchers prefer to know their spot (though they also knew their spot the other way, just not when it might occur). Also, the current bullpen has some good depth and balance, but Duran is just clearly the best pitcher in the pen right now, against batters from both sides. It would be pretty hard to justify sending out Alvarado or Mayza against a LHB pocket over him, or to claim you liked Keller in a match-up better.
Duran's on a nice run right now. The dings will probably come eventually, as they have almost every year for him (and any closer. Well, except Lidge in 2008).