Is this at all a cultural thing? Some cultures still keep birthing to women and the men wait.
I asked my DH to stay up by my head, to hold my hand, etc. He got to cut the cord, which was huge for him and he got to be one of the first people to hold her. I did kick my mom out, and she waited in the hall til DD was born. It was just me, DH, the OB (who I had met just a few days before, she happened to be on call that day) and the nurse whose shift I was on. I am a fairly private person most of the time, but 1. this was my husband, who I was naked with to create this child, and whose support I needed, and 2. there's a lot that you can't control when you have a baby. Giving birth is kind of like having a plan and realizing it may not go that way at all and keeping your eye on the end prize. I won't go into details here, but you should go to a birthing class and find out the nitty gritty details of what you may or may not be able to do or what may occur. I personally chose an epidural and could not walk right for a few hours because one leg would not wake up. I needed a lot of help. My sister nearly died before her first emergency c-section. Absolutely her DH wanted to be by her side.
I think you also need to have some sit-downs with him and talk to him about how HE would feel about supporting you and being involved and being left out the day of the delivery. Is this important to HIM? Does he dream of cutting the cord, being there for baby's first bath (often done while you are being cleaned up), etc? Is he trying to be a good dad by coming with you? Will you nurse? If so, will you not nurse around him? Or will you compromise with a cover? Would it be better for you to be in a birthing center vs hospital? Would that increase your ease? If you wouldn't have him with you, would you have any support at all? Your mom? A doula?
I think that your phrasing of "side show exhibit" when referring to birthing your child in front of the man who created it is concerning. You may also want to talk to someone to help you get to the bottom of why you feel like your husband's presence will make you feel so badly. Your OB may be a good person to ask for a resource. Why do you feel like an exhibit on display even now? Are you reading into his natural curiosity or is he treating you poorly? The more I think about it, the more I feel like this is not just about privacy, but unease with your body, your relationship, motherhood or all of the above. Those are things you should try to address before the arrival of a child who will need so much from you, no matter how tired, stinky, stressed or gross you are (and babies are tired, stinky, and gross, too...you will likely find yourself discussing bodily fluids from your child more than you ever wanted to, out of necessity). If you are not comfortable with an internal exam in the OB's office with him there, you can ask him to step out. Do you discuss your unease with him? Your fears? If not, why not?
I do not regret DH being with me, even if I am an overall private person who discusses only the very important things and doesn't indulge in TMI. My DH is basically in awe of what women can do and nothing about the experience has dampened his feelings for me.
ETA: If he gets squeamish, then he can step out. He would not be the first or last father to faint or need to sit down. But I think it is important to allow him to choose and continue to communicate vs assume on either end. I also like the compromise someone else posted about being draped. Would that make you feel better?