Personally, I thought the problem doesn't seem to be too much red,
but rather lack of blue and green. In Mode Ia, the lack of blue and
green makes all the red into something like (255,0,0), so you just
get a red blob instead of any detail.
When changing to Mode II, stronger blue and green are introduced,
and you get the correct shade of red. But at the same time, other
colors get changed too...
Looks like Mode Ia, IIIa conversions have different models for "red" than Mode II/aRGB.
Here's an image (D50) with the original JPG (Mode IIIa sRGB). Below it are the same image converted from raw via Adobe Camera Raw into aRGB, sRGB colorspaces.
Differences (not-leveled, layer difference) on the right.
Looks like a lot of the texture/"detail" is lost by losing so much of the green/blue data.
Looks like Mode Ia, IIIa are models for color conversion into sRGB space [the sRGB tag is misleading per other articles people have posted; is space for color; seperate from the mode which dictates way colors are process into JPG]. For red, it looks like they tend to blow out the red and drop green and blue (at least in reddish colors).
ACR seems to have the same model for converting raw into aRGB and sRGB spaces. [Difference between the lower left two images is nada]. Of course, the deltas could be difference between ACR and the Nikon bodies (but based on the samples posted, that sounds unlikely). A better control would be to see the same thing done for a conversion via Nikon Capture which supposedly applies the same algorithms for conversion of raw to jpeg in Mode I,Ia,II,III, and IIIa as the camera.
It'll also be interesting to see if the D200 (mode I, Mode III) shows the same issue. Note: the D80 definitely shows the same behavior, so it's not the electronic shutter.
Yet another reason to postprocess (either shoot mode II or raw around reds), sigh.