In OP’s perpective, placing NIN where it is is entirely based on Closer, and nothing else. Hurt could be sad without being horny.

It is possible to view Closer as horny and happy. The infactucation bringing him closer to god, is a successful antidote to the sadness he expresses them curing. Also the horniness infatuation in itself is a pursuit of happiness. IMO, if you see the cure for your sadness right before you, you cannot be/no longer sad. So, as it resonates to me, closer is happy and horny, not sad, despite lyrical evidence of sadness.