I'd be more than fine with a movie that took that kind of bleakness and ran with it convincingly, but making the protagonist yet another generic dude who has to overcome adversity and save his generic wife and child probably isn't the best way to achieve that. In Jaws, we cared about Chief Brody getting back to his family because of the wonderful character moments early on, like the dinner scene. Murtaugh saying "I'm too old for this shit" was at least mildly amusing because Danny Glover's charm and the great chemistry he had with Mel Gibson breathed life into him and made us care. There's quite simply no reason aside from a lazy, empty script that Godzilla couldn't have given us at least some memorable character moments in a "mere" monster movie.
If the point was to indeed make a post-human blockbuster (and I don't think this script had enough on its mind to aspire to that), then don't make a guy getting back to his wife and kid the focal point of the human side.
I'm not trying to be a nitpicking ass..I know it's just a summer movie, but I'm tired of that being such an easy excuse for mediocrity in these movies. Really just trying to figure out why it left me so cold, because I so wanted to like it.