Consider this: the institution is now called "Coast Hospice" -- slightly different mission than before.
Also, a birth CERTFICATE would be easier to lose than a birth REGISTER. (or easier for Rahmbo to lose it for them). It is a single piece of paper stored in a file box someplace, unless it was reduced to a spool of microfilm (in which case, it would be sequentially placed in the midst of other certificates).
So the lack of a birth certificate at a hospital wouldn't bother me as much as the lack of a birth REGISTER maintained by the government.
FWIW / YMMV