I went to new Orleans and brought my 7 months old babies birth certificate but I ended up not needing it
I flew from Oklahoma to California 2x with my son under the age of 1, once at 6 months and once at 11 months- and needed the birth certificate to present at check in. They asked every time.
I’ve never been asked but have brought to every time just in case. Technically they are supposed to require documentation and not a copy in your phone and sometimes you get a rule follower and sometimes you don’t.
I take them just in case. About half the time I have to show TSA. If you don't have it, some TSA won't let you through. So it safer just to have them with you.
Just take a picture of important things