出生證明??? chinese embassies never ask you for that. registering 出生證明 for babies is something relatively new in china. in the west, it has long, long history because of religion of christianity (registering new born babies with churches). it turned out to be a very important aspect of social development.
finally china started catching up in the respect. but i don't think average people getting there yet. my parents don't even know their exact birthdays. my relatives usually don't keep their kids' birth certificates (don't think those are important)..