Great question, Peter. I'm no expert on music theory, but suspect that the only functional criterion is what it sounds like to the listener. As you note (sorry, I couldn't resist) the signature note for a minor, diminished, augmented, etc., must be played by somebody so the listener can hear and (hopefully) appreciate it. I suspect that most listners don't perceive most inversions, but their attention can be pretty easily captured by octave jumps within a chord. Sometimes, it may be more important that a note not be "wrong" as opposed to necessarily being "right." Well, enough of speculation. Time to step aside and wait for those better trained in theory to jump in.