Games shouldn't be replayed, nor will they, absent proof of corruption, as someone else mentioned. In fact, these distractions - chasing rabbits - draw attention from what could and should be done.
Officiating must improve, and that's not all on the on-field officials. Rules interpretation has become so complex. Everything can't be defined to the nth degree. Did his shoulder turn? Did his head turn? Did the ball move? Officials don't have microscopes.
That said, people keep referring to "a lot of other bad calls" in taking the position the Saints weren't screwed. But think about it - that's the defense? Well they were terrible the entire game. The Saints just happened to be the victims of the final botch.
Surely the NFL can and should do better than that. Whether in rules simplification, accountability, training, technology, etc., the league must demand improvement. Fans can help by demanding improvement.
The non-call of PI on the play was so absurd as to be unfathomable, except to conclude the officials simply didn't want to make that call at that time. Is that an individual making a decision, or does it result from guidance he's been given?
I generally like games in which the officials refrain from calling every ticky tack foul. But you can't have a circumstance where players can take advantage by committing egregious fouls because they understand the officials are inclined to swallow the whistle.