But if your offense is averaging 6.0 per carry, the defense will bring up more guys to stop the run which will in turn, improve your passing game (assuming you have a QB who can play well enough to take advantage of the decreased coverage by the defense). So the effective passing game fits your criteria (be able to pass well) but it does so in part because the good run game took pressure off of the passing game.
In addition to that, your defense has likely been helped by time of possession and isn't as fatigued when playing, so there would be a increase in the pass defense, which again is the criteria that indicates the winning team.
So, while it is indeed the success or failure in the pass game that indicates who is successful, the run game still could play a big part in the eventual stat that says the run game isn't important.
In other words... a good running game doesn't guarantee a good passing game or good pass defense but it should improve your chances in both your passing offense and your passing defense.
Do you think the above is false or are we talking in two different directions here?
Don't beat your head against the wall trying to argue this point. There is a group of people that believe that the only thing that matters is "the team that passes more effectively wins". They will just keep repeating the statistic that passing effectively correlates to wins.
The reality is that they just don't have the ability to show statistically how the running game contributes to winning. A lack of ability to show the importance of the running game doesn't not prove that the running game is minimally important to winning.
Simplistic statistics like passing effectively correlates to winning are single variable statistics. Winning is a multivariable equation. The running game and passing game are interdependent variables, not independent variables. The passing correlates to winning statistics can't show which team's passing game improved because the defense played with a defend the run first style and used 8 or 9 men in the box to stop the run. Defenses adjust to a good running game which will always limit the running game statistics. If somebody really wanted to be accurate, they would need statistics that show things like how often the defense played run 1st, used more men in the box, used a read and react style by the DEs that limited the pass rush against teams with a strong running game, etc.. Obviously, some of those things are basically impossible to statistically quantify. You can watch a game and see some DEs hold up on their pass rush in games against offenses with at strong running threat, but how would you ever put that into a statistical format?
Obviously, passing might be more important than running if you could put a number on it like 60-40, but it's definitely not 100-0 in favor of passing. Teams average around 300 yards passing and 100 yards rushing, so statistically passing is always going to "look" more important from a statistical viewpoint.
I can tell you from past experience that the people that believe this concept are not willing to consider that they might be wrong. It's not as-if I'm an old-school, gut-feel, we've always done it this way person in terms of my football viewpoint that eschews some newfangled statistical approach to the game. One of my degrees in Mathematics, so it's not as-if I'm anti-statistical analysis.
If the running game really didn't matter and it was all about passing, then defenses would replace linebackers with cornerbacks. There would be no such thing as a pass rushing specialist that is a part time player because he is poor run defender. The best pass rushers would always play and run defenders would be cut from the team.