Closer strategy depends entirely on the league your in, the rules, the number of teams, etc, etc...
There is no right closer strategy. And, each draft is different.
If my league requires 4 rp, I'm figuring out what closers I definitely want to target, and which I want to avoid. I do not want a team with 2 of the top 3-4 closers (preseason rankings), but I do want one top closer for sure - and I am always targetting Kimbrel or Chapman. Their strikeout totals are 2-3x what some other closers are, their ratios are significantly better, and I would rather have a guy that can get me stats every week then having to throw out marginal sp's.
Totally agree on the over-love for quite a few closers out there. Grili is 36 ffs. Sure he had good k ratios last year and going back a few years, but those are not in highly leveraged, game ending situations. He has not shown he has the closers mentality. Sure he is given opportunity, but let someone else take a chance on him. Same with Wilhemson. Dude was getting lit up in double a just two short years ago.
Then you have the closers that walk too many people. Avoid. It's quite a bit easier to pitch and get k's and saves, when no one is on base. Nothing better than seeing John Axford come in and walk the first two batters and then see him with a 40 pitch 9th inning. Avoid.
I don't want to be on the end of a closer run either, which is why I usually jump a bit early. If there is a closer run, I'll target another area and wait for end game.
I'm not afraid of middle relievers either on my team. Sure, in shallow 10 man leagues, you usually don't want most of these guys, but when you're in deeper leagues, I'd much prefer having a few middle relievers than marginal starters on my team. Kenley Jansen is a great example, but in leagues past, I have had very good success with guys like JP Howell, Scot Shields, Johnny Venters, tons of middle relievers offer outstanding value - plus in an unsettled situation, they can grab a closer job too. You get a middle reliever with a .80 whip, sub 2.5 era, 6-8 wins, and 8-10 saves and 100 k's, he's doing some good work for your team out there. Much better work than Ubaldo Jiminez with a 4.70 ERA, 1.50 whip, 180ks and 12 wins.
I try keeping my ratios as low as possible, try squeezing as many k's per inning pitched as possible and let wins and saves work themselves out. Let other teams fight for the most wins and most saves in the league, I'll be happy to come in top 4 in each of those, and win era, whip and top 5 in k's. With this strategy, you will usually find yourself with a stronger offense too, because you're not wasting mid round picks chasing wins with marginal starters.
I dunno, too much to type and I'm sure I'm rambling - but, in short, I've almost always found that other owners severely undervalue closers and stolen base guys, and I like both of these.