I said more likely, not guaranteed, and of course it depends on the program. There's almost no field where tenure track isn't insanely hard to get, but I certainly know a Fulbright would have helped me get into a better program than the one I'm in now. Patrially because of the prestige, but more likely because of how the experience would have shaped me. As you said, it all depends on the field. If weighing the two options, one should see if deferment of the program is possible for the Fulbright or if a Fulbright would benefit them in reapplying if needed.