Like a lot of situations, I think this is one of those "Yes, they can do this. Yes, their family has every right to hurt feelings because of it" scenarios. The couple can delay their announcement but they have to be prepared not everyone may take it well.
I have to be honest, for me I'd rather have the announcement of my child's healthy birth be a joyous, wonderful experience for everyone, and then deal with the fallout of having to say "No, thank you" to kind offers. It sounds like this couple overall has a loving family which is a wonderful blessing to new parents. I'd err on the side of letting them in, rather than keeping them out.