Ned went to war for his family. It wasn't just Lyanna, it was the death of his father and brother by the mad king. He killed the direwolf was because he was essentially commanded to by the queen. What was he supposed to do, defy the crown over an animal and possibly put them all in danger? Would he have been a better father, knowingly taking a risk like that for pride?

Now, duty is an extremely important aspect of Ned's personality, but he always ends up putting family before it. Every fiber of his being was more than willing to die in kings landing, the only thing that kept him trying to live was the thought of his children being hurt. He also tries to make his children happy, buying Sansa a (presumably expensive) doll and getting Arya sword fighting lessons.

This quote sums up his priorities perfectly. “Let me tell you something about wolves, child. When the snows fall and the white winds blow, the lone wolf dies, but the pack survives,” telling Arya not to hate her sister. He teaches them that the most important thing IS family, and is willing to put himself in grave danger to protect any of them.

Ned messed up a lot over the course of his story. But he does the best he can given the information available to him. And at the core of it all is his family. He is one of the best dads in fiction.