Pulling cables (CAT5e) Vertically with Horizontal Studs in the Wall
Given that the  "horizontal studs" are fireblocking, you should be applying firestop caulk after you run the wires, which favors access holes and drywall repair. So does the possible/likely presence of power wiring in the wall, and possibly other things like pipes. Blindly poking a long drill bit into the unknown can get very interesting, in the unpleasant sense of "interesting."

Edit to add: Also - choose conduit. Cat5e is already obsolete, even if it's "good enough" for what you currently want. Conduit makes it easy to install the next thing when needed.