I go back and forth on Billy Wagner.
Baseball's a game for romantics. And the Hall of Fame is a hall of FAME. For me, the Hall can both recognize, and confer, fame as conditions dictate. In other words, I'm completely comfortable in throwing away some analytics, and indeed being downright inconsistent in who I want to call famous and deserving of fame, and who I don't.
The post-season is one factor, helpful but not strictly necessary, that helps establish the fame needed for induction.
Most players and most pitchers establish their greatness across a very large body of work. Relievers are a special breed: almost always failed starters, they establish their value in inherently small samples of work. If they do great in the regular season but then let their team down when the chips are down in the playoffs, that (for me) is a much bigger detraction to their HoF case than for other players.
To the argument that the playoffs are different and the competition is just better, go look at Mariano Rivera's post-season stats. His makes for an easy Hall of Fame case. Few humans can rise to that level. But I'm not willing to give anyone a pass for not doing the job - they prepare their entire careers for those few moments, and they need to come through, again and again, to be considered the best of the best.
There's no question Billy Wagner came through over and over and over again in the regular season. Still, the reduced workload of a closer makes me unwilling to complain if he never makes it in. Lots of players in the mythical Hall of Very Good have that kind of description.
And Nathan's case is just a smidgen less strong than Wagner's, and then a few other relievers have cases just a smidgen less than than Nathan's. There starts to be a Harold Baines type of slippery slope.
I go back and forth.