I've never carried birth certificates for my kids and never been asked for ID for them, but my DS (16) carries his license in case he would be asked.
A birth certificate is only necessary if you have a lap baby and need to prove the child's age.
I just flew a couple of weeks ago with my 4 year old and they did ask for the birth certificate. I flew on South West if that helps.
The last I knew everyone 16 and over needed picture id. A few years ago we had to rush to get DD a passport because she was turning 16 at grandpas.
I have never taken birth certificates for anyone.
I have never had to show birth certificates for my boys but I did always bring them just in case.
This seems to be a common misconception. Children under 18 do not need any kind of ID to fly domestically. At all. Ever. If they are old enough to talk, the TSA agent might ask them their name to see if it matches the ticket.