It used to drive me bats when people would come to the city where I worked asking for copies of B/C, yes even out of state ones. I don’t know how it works in other states but in Florida the counties and municipalities do not keep copies of birth certificates. A copy can be obtained from a local branch of the State Department of Health as they have computer access to those records. A copy can also be obtained from the Bureau of Vital Statistics. But your run of the mill county or city hall office. No.
I was a county clerk in TX and filed BCs in the county records every day.