no they wouldn’t. but you would probably need to bring your marriage license/proof of name change to anything that involves needing your ID.
I didn’t need a new birth certificate for myself - I use my original birth certificate and marriage certificate which lists my old and new name whenever I need those sort of docs. Not sure if other places do it differently though
It’s always your maiden name on their birth certificate
I changed my name recently but I didn’t get a new birth paper for my daughter so I still have my maiden name on there