I agree with your comment that it wasn't obvious that interceptions would win the game for the Patriots, but I will disagree your use of this:
"New England was the 4th Best Run Defense, but 17th in Defense vs. Opponent's Passing Yards/Game."
There is some kind of saying about "such-and-such conclusion based on a faulty premise." Fezzik would probably know the actual wording better than I, but it applies here.
It's common for teams with winning records to be ranked high against the rush and low against the pass. They are often leading, and their opponents tend to abandon the run and pass on every down. It's not that their pass defenses are so weak, it's that they face so many more passing plays, so the stats can't be taken at face value. Tennessee, Minnesota, Green Bay, Seattle, Kansas City, Carolina, Philadelphia, Cincinnati, and Miami all came into this week giving up more passings yards per game than the 17th-ranked Patriots, and they all have winning records, too.