You can use the color checker directly in darktable!
It’s not super obvious, but I’ll try to communicate it with a couple screenshots with your raw file. (Additionally, I’ve added a video from Aurélien (developer of the module, among other parts of darktable) and the official documentation at the bottom too.)
First, adjust the exposure. For the color checker, go two down from the “black” patch and select the rectangle.
Then, rotate the orientation.
Lastly, visit the color calibration module and expand “calibrate with a color checker”. Line up the guide directly over your image, so all the color patches are in the right place. And generate the profile with your preference to “optimize for”, then run the triangle icon, the refresh icon, and check to apply it to the current image.
Here’s an export:
It’s different from your camera’s JPEG as this is a starting point. To match what cameras do, you’d need to adjust filmic rgb’s white and black adjustments (on the first tab of filmic rgb) and then bump the saturation (just like most cameras are set to do) in color balance rgb’s vibrance preset:
(It’s still different from every camera’s stock output, as every camera maker intentionally skew their colors for a specific style. But most all boost the color and add contrast to enhance the bright and dark parts of the image.)
So, you have a calibrated ColorChecker. Great. That’s not (usually) a real photo you care about. Now what?
Well, you can save the color calibration as a preset in the ≡ menu and apply it for photos with the same lighting condition, if you use that lightning condition a lot. That is, for sunlight conditions (or a place with artificial lighting you visit a lot) or if you had opted to optimize for different types of colors to be more accurate (like the skintones, sky, foliage, and other settings). “None” is neutral and doesn’t give preference to any set of colors.
However, the most useful way (aside from a generic sunlight preset for your camera) is to use this is to snap a pic of your color checker in an area with certain lighting (regardless of the lighting), run through the exposure, rotate, and color calibration steps, and copy/paste (in append mode) from the colorchecker chart to the other images in the same set. Doing this will give you a great starting point with accurate colors.
Aurélien Pierre (the author of the color calibration and many other modules, such as color balance rgb, filmic rgb, and main developer who has been working on the linear rgb pipeline in darktable) made a video on this exact topic, too:
I’d recommend watching it… but it’s super-in-depth, clocking in at just under an hour. If he says anything different from what I’ve said above, defer to him. 
(But I think I have it mainly understood and hopefully have communicated it here well enough.)
If you’d rather read more information, the darktable docs also covers the color checker here: darktable 4.0 user manual - color calibration
Once you have this starting point with accurate colors, you can play with colors in darktable and/or apply a LUT (in darktable) for style.
