I grew up in the 70's and 80's and I thought the same as you at first. I still let my wife know that when she's going through her female time that I'll know from her attitude and I DO NOT want to hear about it or have any other knowledge of it; that's why she has girlfriends.
But, I was there holding one of her legs up watching my first being born and I wouldn't have had it any other way. There's nothing gross or off about seeing that little one come out and know that the first thing that they saw in this world was your face. There is no correlation between the fun hole and the hole that the baby comes out of; they look nothing alike and it snaps right back into place very shortly.
The whole pregnancy you will be excited and anxious and change slightly, but you won't feel the connection that she does, the reality that you are a father. That moment comes when you see that amazing wrinkly little alien baby and you can't stop from smiling so much that it hurts your face; that's when you first feel like a daddy. If you miss that, you will regret it and I'm sure that your wife will make it even worse.