Honestly I think people overuse the word edgy, like they do the word hipster and emo. At times I feel like they just use it to describe modern takes of certain characters that simply doesn't gell with them.
New 52 Superman was far from edgy, and doesn't even come close to the overaggresive fanboy Superboy Prime was. New 52 Superman was dude who tries to find a home for two hamsters belonging to a serial killer he helped put away, helping motivate a kid by giving a speech on how to make the impossible possible and his first action was helping a boy who stole his cape stuck on traintracks.
New 52 Superman was more assertive yeah, but far from edgy.