Just looked at my Original BC and it does not include my weight at birth. It does give the exact time of birth and also has the time my mother was in the hospital before my birth, 1.5 hours.
I would have to go get mine from the lock box at the bank, but I’m pretty sure my old TX BC does have that info (otherwise I don’t think I would know my weight at birth). It may vary by state. Mine is one of those old copies (my very original given to my parents that they gave me when I got married) that is black with white writing. I also have a later copy I had to get for some reason that has an official seal, I think. Been awhile since I looked at that stuff tho. But then, I haven’t run for President recently.