I've two bats that I use regularly. One is a B3 low density cleft. The other a B3 butterfly willow cleft. They both weigh about the same, however, the Butterfly cleft makes a smaller bat as its density is higher than that of the Low Density bat. As an amateur, I hit straight sixes with both, about equally regularly, with similar amounts of effort. More importantly, in this experiment of n=1, I have gained muscle mass in the last two years and now hit the boundary far more regularly than I used to through getting fitter. The only time in the past I used to hit sixes fairly regularly was when I was using a 3lb+ bat.
What point does that prove? Only that the bigger volume bat isn't hitting the ball further, the bigger volume me is.
But lets look further at this, as streaky has done in the past:
http://www.cricketbatblog.com/2015/10/14/the-cricketer-article/Boundaries are definitely smaller and, perhaps most tellingly, pitches make all the difference. You only have to look at the county championship this season with the change in rules relating to who gets to bat first. Pitches are much more benign as a result.
You want to redress the balance between bat and ball? Don't look at the bat. Look at the pitch, the boundaries, the fielding restrictions and the ball. Address these, and if we are still seeing an outlandish number of runs, then maybe you can say "It's the bat".