You consider drag only in relation to the part of the shot played towards the ball. There is only the bat face that acts against the air. This is governed by the laws.
I would like to see you play a cut shot or any late shot really without an initial back swing. Take a tennis racquet, and pretend to drive, then put that racquet inside a sleeve and do the same, now leave the racquet in sleeve and stick a conical sharp card board spine at the back and swing back and forth. You don't have to swing too hard to feel a difference, however minute.
Now as you said the bats may just be too heavy for human mind to register any difference but it doesn't mean it is not there.
Ahh Newton is squirming, twisiting, turning in his grave as we speak.