I definitely follow you up to your last paragraph and it all rings true to me, however I don't quite understand, "It is my opinion that the answer is of the form 'consciousness solves problem X efficiently along dimensions Y and Z' where X is some fundamental component of intelligence, and Y and Z are environmental constraints." Maybe the rest of what I have to say is just because I don't understand the fundamental component or constrains very well.
To me mathematics is the limit of description. I can assign a word to some observable thing and distinguish it from all other observable things. I can draw a picture of it to distinguish it even more precisely. I can use various mathematical techniques to describe it even better, perhaps even to arbitrary degrees of precision. But I fail to see how any mathematical technique can capture --the feeling of-- happiness, pain, etc.. These embodiments can not be fully realized by description alone. They can be pointed to, hinted at, and I think great artists can stir echos of them in other people, but actually experiencing them is beyond the capacity of description. That's why I wonder if experience/consciousness is something fundamental. A subsequent worldview would have as its central concern 'beings' instead of 'objects'; it would not exclude any current or future science, it would just shift it's focus away from abstractions and toward experiential beings -- with conscious beings, which we are, perhaps a special case of a much larger set. The gains would not be material, but perhaps there would be some improvements in the ways we interact with ourselves, each other, and our surroundings.