Since this thread is throwing out solutions, I have another idea - transparency.
Every single week, the NFL should not only review the officiating from all games, but should also publish those results for all to see.
For every game, publish the names of the officiating crews. List all the calls they got wrong, and the official that was responsible. Don't only list the ones where flags were thrown, also list the ones where flags were not thrown but should have been (e.g., the obvious hold on Langford). Over time, you could build a database and see which officials did the best job and which ones did the worst. You could give grades for the overall crew, and grades for individual officials as well. This would create a much larger feedback loop that put pressure on the NFL and the officials themselves to get it right, holding them accountable in a very public way for their mistakes. It would also give the officials incentive to avoid unnecessary scrutiny by putting more pressure on them to get it right and to call games evenhandedly.
Another thing - require ALL officials to meet the press and take questions to explain themselves and their calls/non-calls.
And another thing - lift the absolutely insane prohibition against coaches and players criticizing officials. This prohibition just makes it look like the NFL wants to sweep bad officiating under the rug.
I seriously doubt the NFL would ever do this though, because that diminishes (if not removes outright) their ability to put the thumb on the scales to sway the outcomes the way they want them to go, particularly in very close games.
Of course, someone like Pro Football Focus or another organization could do this ... it wouldn't be as effective as the NFL doing it, but it could embarrass them at times and maybe force them to be a bit more transparent about officiating.