To some extent, I agree with you. But I think over a large sample size those distinctions even out. If you're going to give up a bunch of big plays, your points per play goes up (bad). If you make the offense work for their drives, your points per play goes down (good). And, generally speaking, if you make the offense work harder they're going to score less because of the chances for a big play for the defense (sack, turnover, etc.) or the offense to get a penalty that stops their drive.
Of course, it might also just be as simple as number of plays faced is a pretty available statistic. Not so much on possessions faced.