Idc about me but is this fair to him?

So my baby daddy and I are good friends. We play relationship a lot have been for a year. We do love each other. Not in love tho.He helps me out when I need him and promises to continue to help us out throughout our son’s life. I’m currently 24 weeks. He says he will continue to be in our life in person when he can and video chat when he can’t. To be a father to his son but not not put his name in the birth certificate. It’s like a secret relationship and a secret family. I don’t care to not be with him in marriage. We are good the way we are but is it fair to my son to have a father but be unknown to his paternal side of the family. To not be claimed in the birth certificate by a father that is there with him throughout his life. Isn’t it contradictory. Would I be doing the right thing or am I being a fool to hope that maybe things will change. Would I be a fool to accept this or is it better than nothing. I don’t want either to not have the other in their lives and I know he will be a good father and will love our son and be a good influence. But is it worth the price. In the end is this worse for my son.