I didn't mean to imply that it's not a trait shared by every member of the Batfamily - or, you know, by all the heroes.
She was mature enough to consider hanging up the cape of her own volition several times...and she was always dragged back in the game before she could make that choice. Reckless - oh, yeah she disobeyed Batman's orders order to save his life. You know, like Tim did in order to earn the suit. Disobedient - she wouldn't put up with Bruce's BatGod crap and decided to make her own choices *gasp* independently!
As for badly trained - well, whose fault was that again?
That's an argument Bruce can't win. Either he refuses to take on the kid and orders him/her home, and that means he's an authoritative, self-entitled asshole, either he does accept them and start training them, and that means he's endangering children.
Maybe he didn't entirely subscribe to Bruce's theory that you should allow killers and sickos to run around hurting innocent people without getting put down because the moral high ground is more important.
Hyperbole isn't helping your point here. Bruce has never advocated that 'killers and sickos' should run around. And given that 'the moral high ground' is what keeps him from effectively playing God - him and the other heroes - I find it hard to blame him.