The only reason the other leagues have full time officials and the NFL doesn't is they have games every day of the week throughout their season. You can't easily have another job during your season. That's different with the NFL. There are no other games for them to work during the week. They spend at least 15-20 hours Monday-Friday reviewing game film, training videos, rules quizzes, crew and position teleconferences, etc. In the offseason they are attending camps and clinics often in a teaching capacity to college and high school officials. I agree with bobref I don't think you'll see a significant difference in performance (they are already right 98% of the time) by hiring them to be full time. Some of these guys make more money from their full time gig than their officiating gig (especially the new guys) so they would have to significantly increase the pay to make them all full time. The old saying "the juice probably isn't worth the squeeze" applies. I think they are using some of the full time guys now to put together training videos and other materials. I believe 2 of the guys working the Rams-Saints game are full-time officials.
As for the missed PI call, there are just times where your eyes fool you. That bad of a miss very rarely happens. Maybe a couple times a season. Other "misses" are often very close judgement calls that may only be wrong if you slow it down to watch frame by frame (i.e. some of the catch/no catch calls in the Chiefs-Patriots game). The beauty of football (sports in general) is they aren't perfect. The ball bounces funny. The QB misreads a defender. The receiver slips on his cut. The running back fumbles when he's hit. The coach goes for it on 4th and 2 from his own 28. The defender lines up in the neutral zone. This isn't precision engineering on an airplane. The officials strive to do their very best and the guys working in the NFL are amazing at what they do. But they aren't immune to criticism and it's deserved for that one call.