Yes, the video is right – the whole idea of ‘public’ toilets in NYC is a joke. Sure there are some reasoanble public ones, especially in parks and the NYC Public Library. But those in the subway are a joke and hideously unsanitary. However there is one BIG BIG exception, the newly renovated Port Authority Bus Terminal toilets. Previously a hellhole they are now a vision of sparkling modernity. If you are near there, dash in and be very pleasantly surprised.
Look there are apps etc to help you find your way, but the best general strategy is to know your stores. All the following ones have good toilets: Starbucks, apple, Bloomingdales, Bergdorf and goodman, J Crew, The Gap, Banana Republic, Crate & BArrel, Century 21 and Best buy (but make that the one in Union Square). Sear these names into your head and and steer in the direction of the nearest one if you are in need.
However a word of warning about Starbucks. Their toilets are by far the most popular of all store based toilets, because Starbucks is everywhere. So depending on what time of the day you are attempting to use them, you might have a long wait.