Hospitals may retain copies, but the document in question is a state document, filed by the hospital/doctor/midwife. It would also have anotated upon it, any name changes, official adoptions, etc, and changes back as well. Barak O to Barry S and back to Barak O for example.
Since the state has the document, they can produce it, or a certified copy of it. Not an extract, which is what the CLB amounts to.
The reason for demanding the original is that the certification only establishes citizenship, not the natural-born citizenship required for office. Because adoptive birth certificates and so forth are issued, the mere certification of live birth is not a sufficient credential; the additional detail on the actual certificate is required to dispose of the issue definitively.