The idea with the color checker is to reproduce colors correctly. So, there is no color checker standard. The color checker is simply a card with 16 very well defined colors. The colors are chosen to be somewhat relevant photographically but also to have good metameric properties, what I mean be relatively consistent under different types of illumination.
So you take a picture of the color checker under two illuminations, daylight and incandescent (halogen for instance) and generate a profile that reproduces the colors as well as possible. So, in the ideal case, the color checker would reproduce exactly. Such a profile would be correct, but not necessarily pleasant. What I have seen generally that raw converters exaggerate saturation a bit.
Lars Kjellberg has produced another standard card called QPCard with accompanying software. The QPCard has a different set of color patches that may possibly more suited for portraiture.
Anyway, what I would suggest is to use either Color Checker or QPCard to generate a calibrated color and tweak the colors for pleasantness.
Another post indicated that Capture One is quite sloppy with camera profiles, it seems that they just copied and renamed camera profiles for several cameras. I guess that some of the fault lies with the camera manufacturers, they probably don't give all necessary information to developers of raw processors.
Michael Reichmann and Jeff Schewe have often indicated the need for standardized raw file formats. That would make life much easier for everyone, camera vendors, developers of raw converters and also photographers.
Sorry for the long explanation, but I'm interested in both taking pictures and in the science that allows us to make pictures.
Of course, one could calibrate any camera to a color checker in Lightroom or Camera RAW, but... then I guess the camera would adhere to Color Checker standard, which we may or may not find as pleasing. For my current Leaf back I would not dream of doing so because I find the colors very pleasing at my defaults in Capture One Pro. Colors are critical for an image, and it would seem that ten years into making of dslrs such D800 reddish hues or Canon greens should be things from past?