A club doesn't need to run for lots of yards but running the ball does take pressure off the passing game and does work the possession battle, running the ball effectively that is, which keeps the opposing offense off the field. So yes a club does need to run the ball
somewhat but the league is wired for passing and offense has far more advantages than defenses these days (just ask Ray Lewis). I find that a club minus the all-pro QB (ala Brady, Manning, Brees) needs that running game more so.
The article below, granted over a year old and does not contain the 2012 champion Ravens, goes indepth on rushing and championships. The 2012 Ravens rushing attack ranked 12th in attempts, 11th in yards, t-6th in rushing tds, and 12th in yards per attempt, above their passing rankings (15th attempts, 16th completions, 15 yards, t-18th tds). They had a decent turnover margin (+9) which is a massive key to winning/losing. I believe the stat often touted is something like 70% of winning teams had fewer turnovers in a game.
http://www.coldhardfootballfacts.com/content/ground-pound-mjd-and-how-great-teams-ran-ball/14793/
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Rushing attempts is the key
The Cold, Hard Football Facts, along with other researchers, have often said that you don’t need to run well to win. You just need to
attempt to run often to provide some balance on offense. Former NFL coach Bill Parcells has made the same claim.
This list provides further confirmation of this belief: of the categories above, Super Bowl champs on average ranked highest in rush attempts (7.7th) and lowest in average per attempt (11.6th).
In other words, they largely did not run the ball any better than the average NFL team. But they did run it more often.---