I agree it's worth looking more closely at the relationship between objectivity and subjectivity, especially the mapping from the 'nearest' objective structure we know to various aspects of subjectivity. I think this could be a decent route to finding a brain structure tightly related to conscious experience.
However, my suspicion is that the experience itself will remain not amenable to conceptual analysis—that it will remain fundamentally incapable of systematizing. For instance what would the mathematical description of the experience of a fever chill look like? (Hint: the definition there requires that you don't take a reductive approach to this: it's not asking about a mechanism which produces or is responsible for the experience, it refers to the experience itself.)
That said, I think it would be very valuable to understand better what the limits of our conceptual faculty are: as impressive as cognition is relative to other things we and other animals are able to do, we're being unduly anthropocentric in assuming it to be boundless in capability. More recently I've come to consider concepts as almost like another sense: they are symbolic patterns consistently formed when we are exposed to certain stimuli (like our experience of smells consistently reappearing when exposed to similar molecules). Our conceptual faculty augments the pure pattern-correspondence of our more primitive senses in that the patterns can be associated with other internal patterns, and in that we can generate new patterns purely from existing patterns (using logical and analogical processes), which at some future time may be usefully associated with some never-encountered external stimulus.
Finding the limits to our conceptual faculty is almost synonymous with find its 'structure', which I think would have clear benefits.