I think it all depends on the person and the circumstances of the breakup. Just from reading your situation, I don't think I would.
I really tried to stay friends with my last ex for the same reason: I still see something genuinely good there. But like you, the reason she broke up with me (and the way she did it) made me feel awkward around her, bitter at her even, despite feeling a mixed array of feelings toward her.
As cliche as it sounds, time heals all wounds. I'm open to talking to her again and being friends again, but if she doesn't want to, then I'm fine with it either way.
It takes a long while to achieve such peace and even though I care for her still, I don't care about her at the same time, meaning that I'm content to move on, with the hope I'm moving on to bigger and better things. But Rock is spot on in what he said. In the end, getting over the bitterness and the ill will goes a long way in being fully recovered. Not seeing that person facilitates this.