I had a similar experience to yours while looking for my great-grandfather Solomon Gantcher's death certificate, also after receiving an Ancestry.com hint. It listed his name, his wife's name Pauline, and that of his son Louis, and while it gave the certificate number, there was no date of death. I had always assumed he died in Russia sometime before 1910 and was very surprised to find that he may have died in NYC.
I filled out the NYC Archives order online (leaving out of course the death date) and emailed it with my credit card payment. Within a few days I got a return email with a copy of both sides of the death certificate attached. There was no problem with the missing date for some reason. However, it was not my great-grandfather Solomon's certificate--it was that of his son Louis! When I later checked, I found that another Ancestry.com hint had shown up for my great-grandmother Pauline (Solmon's wife), also with the same certificate number, and this time again for their son Louis. Mystery solved.
It seems Ancestry.com's hints do not in any way say whose death the certificate is for, but just lists the names of the parents, their birthplaces, and the name of their child. There's a separate hint generated for each person on the Ancestry record. This was an unfortunate example of Ancestry's hints sometimes causing more confusion than help.
Great Barrington MA
LIPITZ - Smiela, Cherkasy, Ukraine
DAN - Veliuona, Kaunas, Lithuania
GOLDBERG - Bialystok, Podlaskie. Poland
GANTCHER - Slonim, Grodno, Belarus