For example, the two top teams in yards per running play were the Tennessee Titans (8-8) and the Carolina Panthers (8-8), while two worst running teams per play were the Indianapolis Colts (14-2) and the San Diego Chargers (13-3).
A correlation with number of wins of 0.80 with yards per pass attempt is very high considering that's not even looking at defense or special teams play. In general, you wouldn't expect this high of a correlation because of diminishing returns: if your upcoming opponent has been passing, not running, its way to victory, then you'll try on defense to shut down their passing game at the cost of giving up more yards per run.
Now, A.E. has checked out the last eight NFL seasons, and 2009 turns out to be the extreme case in recent years:
So, passing has been more correlated with winning than running for each of the last eight seasons, but 2009 was definitely the Year of the Quarterback. I found myself writing a lot about NFL quarterbacks in 2009, so at least I was responding to a real phenomenon.
One issue is that there are only 256 regular season NFL games per year, so the sample size isn't enormous, and that's one reason for year-to-year swings.
Of course, when you get to the playoffs in January, especially in outdoor games in northern cities, passing can let you down, such as New England's passing attack getting whomped by Baltimore's running game outdoors in the Boston area in the first round of the playoffs.
A question is whether the NFL's popularity could diminish if the game stays a one-dimensional test of passing skills. Personally, the kind of football I liked best was college football in the late 1960s and 1970s when coaches frequently invented all new offenses (the Veer, the Wishbone, and so forth) and have a number of years of success before defenses would catch up. It was interesting to see teams with wildly different offensive styles on the same field, which you can still see in the college game. In the NFL, in contrast, the skill level has always been so high that gimmicky innovations seldom work.
On the other hand, it could be that fans just like passing more than running -- that the few seconds when the ball is in the air is just more exciting than the ball on the ground. Thus, the long term on-field trend in the NFL toward more skillful execution of passing plays is in the business interests of the NFL. There's worse situations a sports league can be in than that.