A lot of the "turd" headphones require pretty serious boosts and cuts to hit your response target, and the stronger the adjustments are the more distortion you'll get. In addition there is also physics to account for so if the driver is rattling at high volume there isn't really any way to compensate for that.

For an extreme example try processing a set of free/cheap airline earbuds. You'll soon realize that if you crank the bass, all you'll get is mud as the driver simply can't reproduce those sounds. Ditto for the highs, all you'll get are harmonics on lower frequencies. And even if the midrange by design already hits your target curve, it'll still sound horrible as the driver distorts the original signal. Flat frequency response does not equal faithful reproduction!