There will always be people who break the law, agreed, but under anarchy, no one can claim it's their right to do so. That's the difference. Claiming the right to break the law is the essence of government, and that's what is morally repugnant.
I understand what you're saying here, but I think it's irrelevant whether a person claims the right to break the law. The important factor is whether they break it or not, the consequences that follow and the mechanisms for preventing deviant behavior; which I don't feel that anarchy addresses sufficiently.
People are actually primarily cooperative, not competitive. The truly evil are the latter of course, but they're in the minority. Most babies are born with loving generous spirits, it's only their parents who turn them into lost souls.
This is where we fundamentally disagree. I just don't think people are born with loving generous spirits. Have you ever held a hungry baby? There is nothing loving or generous about it. That being said, I think that is where I philosophically disagree with anarchists. I don't trust that the people in my community are that generous and unselfish. I also don't think that a competitive spirit implies evil.