We consider the physical world of image capture to work linearly so ideally we would like our processing to also work linearly. That would be possible if certain conditions were met in practice that however typically are not. For instance CFA spectral sensitivity functions would ideally be linearly related to cone fundamentals in the retina. If they were all we would need is a 3x3 matrix to bop around the needed colorimetric spaces.
Unfortunately that’s not the case: sensor SSFs are just an approximation to the ideal, the problem is overdetermined, there is an infinite number of possibly ‘correct’ matrices, if we want just one the best we can do is come up with a Compromise Color Matrix that minimizes potential errors according to some criteria. Think of it as fitting a curve (or a plane or a solid) to a lot of noisy dots. We sometimes try to compensate for some of the biggest errors via a look up table, but even that is just another form of curve fitting that only really works for the finite number of tones that were corrected manually.
So no, the standard 3x3 matrix cannot be made to work perfectly every time, it can only be made to work ok most of the time. All solutions that look more pleasing are just that, more pleasing. This one time, or perhaps in these types of conditions.
Jack
PS Some additional thoughts on this here.