On first-party controllers (e.g. joycons), you can wake the console by holding down the home button.
(Not sure about the N64-style controllers; I've never actually seen one of those in the wild)
Personally, I count the NES controllers as Joy-Cons, since they slot into the side of the system, as other Joy-Cons do. They don't fit in my Joy-Con charging dock or the Joy-Con grip that came with the system, though.
[1] https://en-americas-support.nintendo.com/app/answers/detail/...
[2] https://en-americas-support.nintendo.com/app/answers/detail/...