Gail H. Marcus
Don't know if I've just been lucky, but I usually find the cemetery staff remarkably helpful. At times, they have gone to their archives and called me back. Or, when they couldn't find something, they suggested alternatives to check (name variants, nearby cemeteries, etc.). More than once, it's helped me identify a relative. Maybe it is the person or the time of day. Or just a lucky chance.
I should, however, note that I've never asked for the congregation. And I've never had anyone offer me the name of a congregation, so I don't know if the real issue is that they don't save this kind of information. And I can say that privacy rules have made them more reluctant to give exact home addresses or names of next of kin, especially for burials in the last 50 years. However, they will verify a name or address if I ask.
But overall, I have been very impressed at how helpful they have been, when, after all, these kinds of questions are a distraction from their main business.