I always travel with my child's birth certificate - no matter where I go abroad. As stated earlier, passports don't show relationship. Even if you didn't need this documentation to transit there could be a whole host of other reasons why having this could be helpful or even necessary. I don't know what the downside would be in carrying the birth certificate (maybe someone can come up with some). We have an extra copy of our child's certificate just for this reason.
Put it another way... You don't need to show ID anymore to enter a SkyClub. If an agent asks to see your ID do you smile and provide it or argue that this level of documentation isn't needed (even though you have a valid ID to show them)? I guess it just depends on which battles you want to fight. When it comes to my kid - I just want to get from A to B as easily as possible while keeping him from raging havoc on everyone in D1.
Carry a birth certificate and this doesn't happen. My 2 cents...