If I print the NOTAMs for a 600nm flight, they're going to typically go 4+ pages of NOTAMs that are fairly specific to my flight (just for departure, along route, and destination airports) and 15+ pages if I print all available NOTAMs in a fairly small font.

No. ATC gets busy enough as is, each plane needs NOTAMs covering their entire route, and if the system is down ATC isn’t getting them either anyways.