This summary is not the rule. The rule text controls.
If someone dies and they were not a U.S. citizen, the public administrator has to tell that person's home country about it. Before the public administrator can get official approval to handle the estate, they must mail a notice to that country's consulate, but only if the consulate has an office in New York City.
This rule exists so the deceased person's home country knows about the death and can look out for its citizen's interests. The consulate might want to check on things like whether family members back home should inherit money or property, or whether someone needs to represent the deceased person's relatives who live in that other country.
For someone dealing with an estate like this, it means there's an extra step before things move forward. The public administrator cannot skip this notice requirement when the deceased person was not a U.S. citizen and their home country has a consulate in New York City.