Hello!
There are a few reasons why you wish to do this before birth:
1) The practical part: indeed, after birth the last thing you wish to do is sort through insurance proposals....
2) If you wish to have a complementary insurance for things like teeth etc, doing it before birth allows you a hassle free entry. If you sign up after birth, then the insurance will ask lots of questions and may exclude certain coverage depending on the baby's helath/birth etc
3) Not having even the basic insurance, there are instance where the isnirance will refuse to pay and claim the disability one should pay and whilst they sort this out, you are left with invoices
4) If you have a complementary and are planning to give birth in a klinik, some insurances require that the baby has the same coverage as you in order to cover the klinik costs
Since there are absolutely no disadvantages to signing up before hand, I would strongly advice you to do so
Simpler is indeed to go with the same company as everyone in the family provided you are happy with their services.
Ciao
C