Through my software development work with DCamProf, Lumariver HDR and RawTherapee I have quite good experience of the DNG and DCP format.
DCP has some limitations by design, and you just discovered one of them.
A DCP cannot alter the color of gray, and as a side effect a DCP cannot be used to correct white balance. ICC does not have this limitation.
The reason is exactly what you describe, the LUT operates in RGB-HSV space and while hue is modified by addition, saturation and value is modified by multiplication, meaning that if saturation is zero you cannot increase it.
Although not a limitation in the LUT itself, the DNG spec also mandates that the LUT entries for S = 0 or V = 0 must be HSV 0,1,1, that is you cannot scale the grayscale axis in value either. In fact, Adobe's own software like Lightroom does not even look at the LUT for the 0/0 entries (I've tested), it just assumes they're 0,1,1 and renders it like that. This makes all of those entries in the LUT redundant.
And no, I don't think this is good design, but it is what it is.