Why is increment more popular than delay? Increment is FIDE's default for their tournaments, although their rules permit delay as well. Most internet chess servers also use increment exclusively.

As I understand it, delay is used by the USCF (for historic reasons?), and, for some reason, by the 2017 Grand Chess Tour.

Why is increment preferred over delay by pretty much everyone but the USCF?
Very simple. Move quickly (faster than the increment) and you end up with more time. Over the course of a dozen or so quickly made moves you can build up enough time to allow for another serious think if required. 

With delay you can never get more time. Once in zeitnot you are permanently in zeitnot.