Well, it's possible, and it certainly sounds like there's reason to suspect some kind of normalization related gotcha.

Maybe the scale of the values, rather than the mean being zero, is the issue?Perhaps a larger or smaller SD would change the outcome? Or a different initialization, especially if using relu. You could also try normalizing the data to have mean 0.1 and SD 1, in case it's some kind of dead relu issue. I'm really spitballing there.