The rule of thumb in Evidence Explained is that you do not list repositories for evidence that is widely available, but that listing repositories is essential for evidence that is available only in one location. I think I agree with the spirit of what Evidence Explained says, but I'm not sure that I totally agree with quite the way Evidence Explained says it. I think the issue is whether something is clearly publicly available or not and if the public location or locations can be easily determined, even if the public availability is only from one location.
Most states or counties, depending on the level of the record, only have a single Vital Records department. But the singleness of the repository does not in my mind mean you have to list the repository if it's clear and in the public domain what the repository is. I think the repository for your mother's birth certificate is very clear and in the public domain - it's the county clerk's office. It doesn't matter whether she got the birth certificate from the county clerk's office and passed it along to you or whether you got it from the county clerk's office yourself. The repository is still the county clerk's office. So you don't need to list a repository.
Evidence Explained goes on to say that you might want to have your own documentation of where you got the record. You would say that you got it from your mother and that as far as you know she got it from the county clerk's office. But that wouldn't help anybody else find the record so it wouldn't appear in your citation. Somebody else would just do a Google search for birth certificates in your state or county and discover the current location of the office in question - which might not even be the same location as the office was when your mother got the birth certificate.