Could we use that data to somehow modify the audio signal to try to remove the harmonic distortion?
The answer is really no, harmonic distortion is a nonlinear phenomenon which basically means you've added something that wasn't there before, instead of just moving the original signal around in time or amplitude.
Frequency response is correctable using EQ, same with phase response.
However, distortion doesn't really work the same way. It adds energy at new frequencies. So to remove THD that occurs from a 1khz input, you need to somehow remove the distortion that pops up at 2, 3, 4, 5 khz. But how to do that without affecting the normal, intended sound at 2-5khz ? Your adjustment would need to vary depending on the input at 1khz. And this is a different and much more complex tech altogether than the EQ or FIR that we are accustomed to. In fact, I am not aware of any tools that do this.
You can minimize distortion using EQ by lowering the signal in areas with high distortion. However, you're also changing the FR that way.
There is no way I know of to remove distortion by itself.