Although that does not look like any birth certificates I have ever gotten from the vital records office (32 so far) the ones I order are a little older.
I agree that if he paid, the state should provide or refund. I once ordered a marriage certificate that they said, in a letter, had a valid number and should be on their film, but it was missing from the film - and they were issuing a refund.
Unless you have the canceled checks and can prove you have complained previously about not receiving it - I don't think there is anything you can do except try Schenectady.