No, Im learning.
If we talk psycoacoustics , how an instruments caracter is defined by the brain/ear, is it good enough with a flat frequency response and good directivity then ?
Yes "but" ...
Yes, frequency and directivity capture all of the first order effects of the instrument character. Distortion (both THD and IMD) can contribute along with compression where the volume increases that are recorded aren't delivered linearly. These characteristics can all be measured, of course, and these play a smaller role than the two items you identified.