My husband and I have created two children and he was in the delivery room for one with me (couldn't be there for the birth of our second unfortunately). He has plenty of rights to our children and he is a wonderful father. However, first of all, my husband was there for me, not to see the birth. In fact, he joked most of the pregnancy that he would rather be outside passing out cigars. And had he reacted badly or behaved toward me inappropriately for the first birth, it wouldn't have mattered about whether he could be in there for the second because I would have asked him to please not be there and barred him if his presence caused me too much stress.
It doesn't matter if it was important to him. What should matter most to him is the health of his child, which during delivery, means worrying about the health of the mother, including how much stress he may be causing her. Boohoo he doesn't get to witness the birth. Many men don't get to do that. Most of them are just plain thankful that they are in their child's life. His reasoning sounds pretty selfish to me.
As for child support, if my husband and I were ever to split up, we would almost certainly share custody and our earning potential is right around the same amount and we will have added relatively the same amount in the future to raising our children, which are key determining factors usually in determining child support. If the courts decided that I deserved child support vice my husband, then I would ensure it went toward my sons, just as I would not have any issue paying him should I be the one making more.