The waste is still collected by truck from the terminal, and presumably driven to somewhere else in New York.
London tries to reduce large vehicle traffic by taking the waste away by barge; you can see the place in central London here [1] (I like the irony of "Smuggler's Way".) The Roosevelt Island waste terminal is at the edge of the island, so this is a missed opportunity -- if there's anywhere to accept it by barge. (London's used to go to [2], a quick search suggests it now goes to [3].)
I live in a neighborhood with this system installed. In fact you can see it at 3:28, pretty much all footage from that point on is from there.
When it works, it REALLY works. There is no trash in the streets, it's all in these tubes that you place your trash in. Some buildings have these tubes integrated, some have the tubes in the street kind of variety. Every now and then you'll see special trucks come in that quite literally suck the garbage out of the ground to empty out the underground storage.
Sometimes it doesn't work though. Maybe it's because people fill up the tubes with weird shit but I think mostly it's just because they don't empty the system often enough. When I first moved in this was a big issue, but now I haven't seen it in a while (actually years probably) so maybe they've figured out the a schedule better.
It is really nice, it works well and truly is an out of sight out of mind kind of thing. I'd imagine it's easier for garbage collection too since it's centralized, you don't have trucks coming around to each house or block.
There is some garbage that doesn't go in the chutes. Electrical appliances, glass, larger items that won't fit such as pots and pans and lighter furniture. They all go in special garbage rooms so still off the streets, but not quite as cool as giant tubes underground.
To your point of "out of sight out of mind" I wonder how much this impacts recycling programs. E.g. I'll just throw everyone in the trash because it's gone vs I'll recycle plastics etc separately which reduces "smelly" garbage.
As to cost https://www.nyserda.ny.gov/-/media/Files/Publications/Resear... suggests it’s 10-25% cheaper to operate, but when you factor in capital costs, it's 40-90% more expensive, if I'm reading the abstract correctly.