The official who oversees voter registration in New York City is the 80-year-old mother of a former congressman. The director of Election Day operations is a close friend of Manhattan’s Republican chairwoman. The head of ballot management is the son of a former Brooklyn Democratic district leader. And the administrative manager is the wife of a City Council member.

As the workings of American democracy have become more complex — with sophisticated technology, early voting and the threat of foreign interference — New York has clung to a century-old system of local election administration that is one of the last vestiges of pure patronage in government, a relic from the era of powerful political clubhouses and Tammany Hall.

Already this year, the New York City Board of Elections failed to mail out many absentee ballots until the day before the primary, disenfranchising voters, and sent erroneous general election ballot packages to many other residents, spreading confusion.

Now, the agency is facing perhaps its biggest challenge yet: a heated general election, during a pandemic, under a president who has fomented distrust in the legitimacy of the vote — including by pointing to the problems in New York as evidence of widespread fraud, an unfounded claim.

It is also the first presidential election in New York with early voting, which began Saturday with tens of thousands of residents flooding polling places.

“I expect the B.O.E. to pull this off — there’s no other option. It’s the most important election of our lifetime,” said Scott Stringer, the city comptroller. “But we shouldn’t have to hold our breath because of their gross incompetence.”