Answering my own question here, as there has been some very helpful input, but I think this might give a more accurate answer to the specific question. There was also extensive dialogue around the same question over in the Darktable forum of pixls.us, which contributed a great deal to exploring this question and colour theory.

In short, the card in question, and similar four-panel cards, are essentially a simplified version of the Kodak colour separation guide and similar tools. This means their function is as a (slightly old-fashioned) printing calibration tool; they are intended to allow you to verify the colour properties of a printed copy of a photo and, if required, make corresponding adjustments. In scientific and research contexts, they're also a useful quick reference for checking that an image is true colour or eyeballing any discolouration.

As for using it to correct white balance, yes it is suitable for that purpose, and it does deliver a drastically improved result - as Dan mentioned - but as explored here there are still significant advantages to using a proper colour calibration solution. Four-panel cards, however, are not that solution.

Speaking of colour calibration solutions, as Zeus points out (better than I can summarise) there is a lot of complex colour science underlying this deceitfully complex question, but unless you want to calibrate by eye (and I thoroughly enjoy doing manual colour processing on my non-academic photography) you're basically tied to solutions like Calibrite's ColorChecker, which account for every link of the chain. As they summed up super neatly:

Manufacturers of the calibrated cards like ColorChecker have instructions (usually in the software) similar to "this cell, under a given lighting conditions, have such-and-such chromaticity. When displayed in sRGB, it should have such-and-such relationship between R/G/B."

I have spotted in my searches a few cards that might take a similar role to a ColorChecker, like this awesome scale card for Australian rock art (below), but given the general lack of information on this web page, my understanding is that cards like this are actually intended for visual Munsell colour assessments - essentially as a context-specific pocket alternative to carrying a full Munsell chart - rather than for any form of digital image correction. If I'm reading their webpage correctly, it's another case of relying on correction by eye if you want to use this for digital image correction.

It seems like there's really no alternative to the immensely robust colour science behind high quality 24- and 48-panel checkers. So to answer my own question, no - there is no workflow to retroactively calibrate these images based on the included reference card other than manual correction.