I think she did, but only because she wanted to be seen as more then the slayers little sister.
I think the reason she cried after xander's speech is not because she wasn't a potential slayer but because he had pretty much proved to her she didn't need to be, she was amazing anyway as she was.
We all know xander is and has always been the one in the group with no powers or all supernatural like and must have at one point felt like dawn was feeling, like the odd one out and stuff but xander stopped willow from ending the world without having powers, so i pretty much think that's what he was saying to her.
She didn't need to be a potential slayer because she was way more then that.