Can the New York Times legally copy Naomi Wu's video without permission?

As Obama used to say, yes they can. The NYT video is amateurish but I don't think their intention was to titillate anybody.

However, Naomi Wu is right about Sarah Jeong (who did not let her incompetence get in the way) and Vice magazine (which was not really interested in Wu's skills as a DIY specialist).

More info in these videos
https://www.youtube.com/watch?v=RiLFaT121zs&t=27m46s
https://www.youtube.com/watch?v=Y0dkwwV_iaw