My 2 cents (pence)...

To me the 5D footage is way over saturated/contrasted. The H1 is more natural but in this instance would probably be helped by an increase in contrast/saturation to darken the lows and reduce the effect of the noise.

Although I use a Z5 (which is excellent in low light, I have to say) I almost always manually expose in these situations (and fix the aperture to avoid ramping). In low contrast situations, upping the contrast can help too. (Can also be done in post).

I don't see this as a good example at all. The woman in the XL-H1 clip is half in shadow and the scene badly lit, then the same woman is well lit (and overexposed) in the second clip.

I'm sure that if the gain settings were way down and the 6X lens was used wide open, this would have painted an entirely different...and cleaner, view of the scene with very little noise. Yes, the 5D will be cleaner, but the H1 would be nowhere near as bad as shown if it were used at the exact same time as the 5D in the same light.