Might just as well shoot him. You now have a couch ornament that stinks and still bites.
Come on now... Neutering him was a good move, along with some strong training that you and many of us have suggested.
Sometimes neutering is a last resort to control certain behavioral aspects, but aggression of the type explained by the poster is not one of them.
This dog sounds like a psyche case. The neutering will help with behavioral things like running the neighborhood, leg humping and other behaviors, but this dog sounds like he is brain damaged. His anti social aggressiveness will not be going away and may well get worse. It would be interesting to test the dog and see if he just needs a good can of whoopass. But that is a matter for his owner.
I hate having to put a dog down, but I hate it even worse when I delayed and it resulted in injuries to a family member.
I would not hesitate with this animal, but you are correct that I may not be seeing the entire picture.
Growling sometimes means little. There are other body language aspects that combined with growling that include ear positions, exposed teeth and general position and stance that indicate a dangerous dog to have in a family setting.
I have owned many aggressive dogs that I have not difficulty controlling, but I keep them away from kids and the like.
Some dogs cannot be controlled and this makes them useless and dangerous in a family environment. A loaded gun with no safety. A ticking bomb.
From what I read here, I would put it down.