NY record search advice

Hello, I'm looking for advice on searching for NY birth records. Background gf father was born in NY doesn't have a birth certificate, ran into issues trying to request one and gave up from frustration. There may have been an issue with names or spelling with one of his parents. What I have found is vital records with NYC and NYS, sending correspondance to see if they can do a record check with limited information name dob.

I've noticed there are other geneology resources, curious if anyone can make a suggestion so I can target those resources, I'd like to avoid registering or setting up an account for something that isn't useful. The purpose is she wants to send her parents on a cruise, but the b certificate is a hinderance. I'd like to see if we can find a record or at least a definate path forward.