In a kinship proceeding, the cousins really have to not only prove who they are, they have to prove the non-existence of prior classes (spouse, children, siblings, etc). The level of proof is high, and the money is not going to be released if its not proven.
A related situation happens when such a person makes a will. For a will to be probated in New York, you must prove to the court that all persons "adversely affected" by the will have been notified. These are the people who would inherit under intestacy. Proof of their notification is shown by their signing a waiver, or by their being served with a Citation. What happens if the decedent made a will, and the distributees are cousins? Do you have to go through a "kinship proceeding", just to figure out who to put on notice for the will? The short answer is NO, you have to show "due diligence", both in searching for these people, and in notifying them. If you have looked for such relatives and can't find them, the court may let you serve them by "publication". That's what those little notices in the newspaper are. Those notices happen after a diligent search has been shown, and the court has authorized service on these people by publication. The notices are basically saying to these people....."Hey, the Will of Joe Blow, your uncle, is being offered for probate on July 15th, show up or forever hold your peace. It's rare that anyone responds to these, though it has happened. How much diligence is "due" diligence? I have found the courts to be practical about this. If the Estate is very large, they expect more diligence. They may require a professional investigation. On a smaller estate, they will sometimes permit the attorneys to show a search using available public records, including the internet.
In the overall scheme of things, a due diligence search in the course of probating a will is a much better situation than laughing heirs and a kinship proceeding. Better for whom, you may ask. Better for the cosmic karma of the universe, something I find myself noticing more as I get older.