There is enough redundancy between portions of the 7th and 8th ave lines to permit partial shutdowns and yet five years in they still can't seem to finish their hurricane repairs.
Is that the official reason though? "The service cannot be stopped even for planned maintenance" doesn't sound like a reasonable constraint.
They have been doing shutdowns of particular lines for planned maintenance, but almost always with replacement service in place (either other nearby subway lines, or for more isolated lines, buses). And then there are major tunnel repairs, which can lead to shutdowns of over a year. (There's an upcoming shutdown on the L which a lot of people are dreading, as there is no good replacement service of anything like the same capacity.)
Which is great but it sounds like they need to compromise. Does New York have night busses?
Yep. In fact they're ridden so infrequently at night, they have a request-a-stop program after late-night hours: