I know the Giant's snapper (deOssie) was a LB at Brown, he actually got drafted. But I think he was drafted as an LB and happened to be good enough as a snapper to hang on - not sure what the norm is. Nice to have another LB on your coverage team though, he can hit.
I think it was Ober wasn't getting down the field fast enough to contribute to the punt coverage, his snaps were fine. But you're right he's undersized, and speed will still be a problem in the NFL. I just don't ever see NFL snappers making tackles or contributing to the kick coverage, so I wonder if they only care about snap speed/accuracy where Ober might be good. There are snappers listed at 235, so depending on whether he's put on weight that could be fine - they can't hit snappers anymore anyway, he doesn't have to block.