If there is no gander to protect the nest, the goose will take over that job.
In my opinion, the goose is much more fierce, but will protect a very small territory around her nest. The gander will protect a very large area.
I do not have any problems with my ganders. I do not treat them like lap pets. I expect them to behave with manners at all times, and they do.
I don't know if a stranger could walk through their pen, but I certainly can, even when the goslings are newly hatched.
I tell you, though. It isn't a good plan for any dog to get close to the goslings. I don't think I would have geese around small children, because children can't assert themselves as flock leaders and geese can be really scary.