Yes, I think child free restaurants might be a good idea for more formal dining. I'm a parent myself and I know I signed up for the extra noise and mess that goes with children but not everybody has and the rest of you deserve to dine in peace sometimes when you're having a night out. Frankly, if I'm out for dinner without my children I would probably prefer not to have to put up with somebody else's kids screaming instead. On the other hand, for less formal dining spaces like cafes I think you might just have to suck it up, especially during the daytime. Being a stay at home mum can be pretty isolating and mums do need somewhere to meet and just have a cuppa and relax sometimes (yes, we could do it in our own homes, but when you're at home all the time you just need to get out once in a while). I agree that parents should try and keep their kids under control in public spaces, but I know how difficult that can be to actually do, even with the best intentions. If you really want to go somewhere there won't be kids you could always have lunch in a pub.
I really can't see anything wrong with a woman showing her ultrasound photo to a friend in the cafe. It may not be for you, but it is a life changing experience so of course she was excited about it and her friends were excited for her.