You would most certainly remember if you were at the birth of someone famous. As they became famous, you would recall a memory, or look up a record, and brag to friends that YOU WERE THERE. You are human aren’t you, and it is entirely human to want to mention that you are part of history, if even in a small way.
If one of the PARENTS were famous? Sure.
If the baby later became famous? Not a chance.
“You would most certainly remember if you were at the birth of someone famous.”
Not if they weren’t famous at the time of their birth.
“As they became famous, you would recall a memory, or look up a record, and brag to friends that YOU WERE THERE.”
You’d have to have the memory to recall, which would have had to have been stored before they were famous.