Ok, I just did that, there was "flat" and "neutral" for Nikon. It makes a huge differnce, but I still don't see the bluish hue in the grey bottom concrete. So I'm suspecting a color shift from the camera, that makes greys warmer than they should be, regardless of white balance. Otherwise the white wall would be to cold too.
Personally, I wouldn't try to reach any conclusions from one image. I would try multiple different profiles/recipes on multiple images, to try to find out which ones might or might not work better for you.
I am going to calibrate my monitor soon (for the first time).
This is really a prerequisite for doing any kind of color evaluation of digital images. You can't really learn anything useful about your camera if your monitor is displaying wonky colors (and it probably is, to some extent). You can verify this for yourself by displaying your test picture on any other screens you have (phone, iPad, TV, whatever). They will very likely look at least somewhat different on every screen.
The Z5 is essentially just as capable of producing accurate colors, if used in a well-managed workflow, as any other consumer digital camera out there. So you can do it with your camera, with some effort.
Couldn't I find a camera with a more neutral color science from the start, if that helped?
I doubt it. Every digital camera comes with multiple color sciences -- i.e. all the different profiles/recipes you can choose for any picture. Each of those profiles produces a different mix of colors for any given image.
At the next step of your workflow -- your editing software -- you have another collection of multiple profiles/recipes, each of which can give your picture different colors. Depending on the software, it's completely possible that none of them exactly match any in-camera profile -- i.e. Capture One's "neutral" may not be the same as Nikon's "neutral".
So there is no single "color science" at play here; there are many, many of them.
The reason for this is that the vast majority of photographers don't want to be limited to only neutral or "accurate" colors; they want creative choices. Plenty of people set their camera to "Vivid" and then boost the saturation even more in their editing software, and they are happy as clams, and good on them for it.
So rather than trying to find a different camera with, say, one particular built-in profile that works better for you, I would try experimenting with what you already have, either with in-camera profiles, or with profiles in your editing software. You may solve your problem with minimal effort.
But if that doesn't work, it is possible to generate a color profile for your specific camera with colorimetric accuracy as your intention, use that profile in your editing software, use a carefully profiled monitor, print on a carefully profiled printer, and manage the color transforms all the way through the process. It takes a fair bit of knowledge, and possibly a modest pile of money, but it can be done with any sophisticated camera like your Z5.
This is what museums do, or clothing and paint catalog photographers. They don't look for a camera that is perfectly accurate in all conditions out of the box (because none exists); they carefully profile and color manage the camera they have, along with the rest of their workflow.
