I've been trying to articulate what bothers me about this all day... and I think I've got a bit closer now. To me, this is like the circles of care when people are sick: you have to comfort in, and dump out: www.girlfriendcircles.com/blog/index.php/2013/04/how-to-respond-to-a-friend-in-crisis/
I feel like having your first child is a Very Big Deal. And that, really, the focus of that experience needs to be 110% on the little nuclear family with a sudden new addition: Mum, Dad (or second Mum!), baby. Basically, whatever it takes to make that family feel happy and adjust to this colossal, amazing, terrifying adjustment to their life needs to happen. They are the priority. They go in the first circle.
I do think that, because birth is so intensely physical and therefore so intensely female as an experience, the woman's wishes need to come first. If she feels comfortable with her own mother there, but not the mother-in-law, that seems fine to me. I see no argument here that there is an injustice in allowing one set of relations to be present over another, if the woman at the heart of it all is more comfortable that way (and I imagine that a lot of women would be a lot more comfortable accompanied by their own mother than their mother-in-law).
Basically, in my view, everyone should put the unit at the heart of things first - and they shouldn't dump 'in' onto them by putting any unnecessary expectations or demands on them. I feel like underlying many of these AIBUs about visiting and birth is a tussle about social priority between the adults, which has very little to do with the interests of the child or the family (not saying you are like this, OP, it's more of a general comment about threads like this, which recur regularly). In other words, there is a kind of competition about when X or Y gets to see the child and how that compares to others - it's kind of a weird pecking order. Short of a couple inviting absolutely everyone to witness the birth itself, I can't see how the couple at the centre of it all can avoid offending someone! So I'm not surprised some are saying 'no visitors until we know we are coping' instead of dealing with that pressure.
I realise that the experience of actually giving birth is a bit different from visiting, but I think the same rough principles apply. More and more couples these days are cautious about how they will react and want space and time to play it their way and to have time and space to experience it together. It sounds like the OP's family are quite close, but in many other cases, the bonds of family are loosening somewhat - if you live miles away from people, and they're not going to play a huge part in the child's life in terms of caregiving, then I don't think there is the same sense that the child is born into a family-community or the same urgency to 'introduce them'. (The community may well be that of friends instead).