MOST of the protection trained dogs that I know are very stable and well-rounded dogs. I have seen them get mugged by kids as soon as they come off of the field, seen them stand by their owner's side in a huge crowd of people, and have seen 100s of dog aggressive dogs get together in one spot and not seen a single fight. I actually have yet to see one dog that I would consider unstable at any working venue, seminar, or training club that I have ever been to.
As for my pup, I have done little obedience with him and see no reason to put him near small children until he has more control on him. He's actually quite the social butterfly - loves everybody and even kisses the vet (it actually kind of annoys me as I typically appreciate a more reserved dog). We play and work near kids playing football or riding around on bikes or kicking soccer balls, etc in my neighborhood and at the parks in my area or at the beach. That is working on keeping his focus on his handler versus wanting to run after the little people. I also do sometimes allow children who are old enough to understand and follow directions come up and pet Cairo. For all I know, Cairo could be awesome around babies but it's not a chance I am willing to take at this stage in the game. I fully anticipate that when he is older and his brain has settled a bit that we will have no problem being around younger children.
As for handler aggression, I have had a couple dogs come up the lead at me and it is never a real bite. It is a warning, a way of communicating something to the handler; it can also be a way for the dog to try and continue the game. I don't mind this sharpness at all and have generally found that a firm but calm command for the dog to sit or heel ends the entire episode. In fact, I would label very few "handler aggressive" dogs as being truly aggressive toward their handler.
Not all dogs are suitable for all people. There are plenty of working dogs that are totally fine in any situation and then there are some dogs that are more of a liability. But if that is the kind of dog that someone wants to own and work with then that is their right, as long as the dog is contained and controlled appropriately.