^I think that's how they interpreted it with the animated shows. Despite this they initially played her as being very naive and sheltered, so one can hardly say she'd already be a hardened veteran right out of the gate. Oh and they also implied that her father was Hades, which IMO is a lot more interesting than it being Zeus. I mean how many dozens of demi-god children did Zeus father for goodness sake?
Being a daughter of Zeus would better explain her abilities than being a daughter of Hades. She is strong like Hercules and can fly, she doesn't command the dead.