Two shots using the 5DIII and 1DX. Both shot at ISO 1000. Both using a 600mm F4 IS with 1.4xIII extender. Both days were taken with complete overcast against a less than pleasing sky...Normally I would pass on any attempts in this condition but it makes for good comparisons in bad lighting. No post filters except 25 sharpness on both from the RAW files. 5DIII was at f/8 and 1DX was at F/10.
BE8 is the 1DX
NL7 is the 5DIII
Note that the 5DIII has smaller grain pattern than the 1DX due to pixel density. Chroma noise is about the same, I think maybe less with the 5dIII but I could be persuaded that it's the same. For those that say the 1DX has no chroma noise you are wrong. It may have less at ISO 25000 but you cant say it has none. Post with CS5 we can process out chroma noise pretty much at all ISO levels with both cameras.
Crops are 100% at 300dpi
(FYI, dpi only matters for print. On screen, you only have image dimensions in pixels and the PPI of the screen. It really wouldn't matter what dpi setting you used, on my screen it will always display at 103ppi. The average desktop screen these days has around 96ppi, most phone screens are 250-330ppi, and most tablets are 140-250ppi. People will see the image at the resolution of their device, regardless of what setting you choose when exporting. If you want everyone to see it at certain dimensions, there really isn't any way via the web, although I would imagine that if we actually could see those crops at 300ppi (my Lumia has a 336ppi resolution), they would look far less noisy than they do.)
Any chance you could post screenshots of the original histograms for each of those photos (without any post-process edits)? Given how much noise you have in those shots, and given the overall tone, I'd wager that a higher ISO setting with a greater amount of ETTR would produce better results from both cameras. I recently read some interesting info on exposure from one of my favorite bird photographers...Art Morris, who is a staunch advocate of ALWAYS exposing such that the rightmost ends of the histogram end up half way into the rightmost vertical box on the histogram display. Since I started doing that, my photos have been less noisy after post-process exposure correction (which, actually, tends to be minimal most of the time). Given what I've learned the last several months with my 7D, I would be willing to bet both the 5D III and 1D X could produce less noisy exposures for the same images you posted if you used both a higher ISO setting, and dropped the right-hand end of the histogram half way into the rightmost box.
It may not seem like "getting the exposure right in-camera", and if one was using film it would indeed be an entirely incorrect way to expose. But to maximize the potential of digital equipment, we have to think about it and use it differently than we think about and use film. I'm a staunch believer of ETTR, or expose to the right, when using modern digital cameras, and in my experience it does result in better dynamic range and noise performance out of any digital camera.