I posted a fairly long bit on this not terribly long ago on these forums, I think. I used to be in the camp that said closers were silly, but more and more I disagree with it. I think most pitchers are most comfortable when they have a reasonably well-defined role or usage pattern. A "fireman" role makes it a lot more difficult to get into a comfort zone. If you might come in sometime in the 8th or 9th, but know to start warming up after the 7th, I can see that. But if you have to be ready on short warmup starting in the 6th inning, that really changes your preparation both mentally and physically. For the same reason, I prefer to see as few mid-inning pitching changes as possible, at least without advance planning. I want to see my team's relievers warm up at their own pace, not hurried into games. It helps them reach their maximum effectiveness, and it helps a little with injury risk.
Of course, all of this depends quite a bit on the overall bullpen situation. If I'm the 2015 Mets, maybe I bring my closer in early in more situations (as they in fact did, at least relative to most MLB teams). That's because my next-best relievers are a huge dropoff. But if I'm the 2016 Yankees, I don't NEED to use Chapman in the 6th, because I have 2-3 other elite arms that can handle that situation. If there isn't a huge talent gap, then I want to maximize the comfort level - and ostensibly, in turn, productivity - of each of my pitchers individually. As an O's fan, I'm perfectly comfortable with Darren O'Day putting out the fires in the 7th and 8th, or Brad Brach in the 6th. Those guys are good pitchers.