I once wrote a camera profiler for my P45+. From a profile computed on 4 patches, I had no difficulty in reproducing a colorchecker, color rendering was -mathematically- superb
The digital back guys claim that the low native ISO of their backs is partly due to very orthogonal color filters that lower the amount of light passed through to each pixel - conversely, a monochrome back gets all the light on every pixel and cannot discriminate colors. The dSLR guys are thought to often have made a compromise which improves ISO at the detriment of color discrimination. Of course, mathematically you could recover the information from less orthogonal filters but obviously the less the orthogonality, the less color discrimination you will have given that you only have a finite precision to work with.
The only practical color issues I have seen with Kodak CCD's (Phase, Leica) are the "magenta face" issue where skin on some very white eg. red-haired people takes on a magenta cast. I was never able to nail this down precisely, but I think it has something to do with out of visible roll-off with the camera seeing "under" the skin, and the only solution is to edit the profile by hand, or the image. My Nikon D4 also suffers form this to an extreme degree.
The MF guys have really done a lot of work to bring us good color; this is not only the chip suppliers eg. Kodak and Dalsa,, as the software, C1 especially has much better color editing controls than Lightroom, and very good rendering.
Incidentally, I have a suspicion that the reason color editing is so kludgy is because of software patents - namely all the "good" interfaces that operate in intuitive color spaces have been patented. An interesting consequence of the US patent system is that most simple and effective solutions are visited by the first parties to explore a market, get patented, and as a consequence are never widely deployed. A good example are the Kodak patents on profile editing which have essentially made it a certainty that there will be no good profile editors on the market. There is a similar situation with display calibration, I believe, with patents essentially blocking all the cheap and cheerful solutions which anyone here could "invent" in a few minutes. A consequence is that display calibration has remained a luxury, and as a result most internet users see color that has no relationship to the original imagery. I once designed and fabricated a calibrator with a component BOM of $2 and I could easily do $1, but I couldn't sell it in the US.