Bart, thank you for your prompt reply!
Even if the card is neutral, and that's an 'if', ambient light and thus local colored objects will change the reflected colors, even across a spectrally neutral reflector. [...]
Whatever card I use, and whether I use camera Raw or C1, as mentioned, I get very different results, mainly, and supposedly because of the reasons you bring. Only, I want to remain focused on the practical side of things. I am working in a tethered studio setup for portraits. As I charge a quite some money, my costumers become very big specialists in detecting clothing, hair and skin tone issues. And they do have good eyes. So variation in accuracy of +- 400K and +-20 in tint, is huge!
Is the only solution to this just to click around and playing in this area of +-400K and +-20 in tint until I find something satisfying? (I will later warm the picture up a little most of the time, but that's an other story)
In C1, I crop the image down to the whole gray/neutral card area (mostly the classic color checker) and apply one of the two Auto white balance possibilities. All of which give me very different results, not just in numbers, in looks. 6200K/3.1 (picker), 6044K/1.9 (Auto), 5891K/3.1 (Mode "auto"). It is mot often this last lowest number, which appears the most accurate.
For example, a picture taken in a studio painted with neutral gray, where there is nothing else except a neutral gray table, so there is nothing that the gray card could reflect, and still big differences.