Perhaps it's different for men, but there are few events that I would experience as more disfiguring than losing my hair, and they all involve severe trauma. In fact, I have a facial scar from a fall when I was a kid and I don't ever think about it. But one time I managed to sear my hair off with too much bleaching and the couple of months until I was sure it was growing back were some of the hardest I've ever been through.
Is that socially constructed? I suppose having a nose (rather than just a pair of nostrils) is socially constructed. But we live in a socially constructed world and some things are harmful, even if they have no real substance.
Edit: but I'm proud of you if you can get over some of those things and more power to you. A handsome man is a handsome man, hair or not. I'm making a comment about how this sort of therapy can be beneficial to women, also, for whom losing hair is not "natural" (in many ways).