One thing to start is to understand is that White Balance and accurate color reproduction are two different animals.
For example, if you put a WhiBal card under a CFL source, you can use auto WB setting in camera or in post processing to get a neutral gray rendition to the card...correct White Balance. But if you have a photo of the ColorChecker in the same shot, you might still notice that some of the color patches are not truly accurate in rendition, due to gaps in the color spectrum emitted by the CFL.
This isn't a really tightly controlled example, but it does illustrate the principle in the above paragraph.
First Daylight (cloudy day) balanced ...
Next, CFL balanced with Lightroom...
Note that although both are White Balanced, the reproduction of colors in the CFL shot is sometimes 'the same' and sometimes 'quite different'.
And even compare the difference in color rendition between a 'Daylight CFL' and an ordinary CFL, both White Balanced shots.
Daylight CFL
Ordinary CFL
The ColorChecker came about in the days of film, because although a dozen film emulsions might all be 'daylight balanced' emulsions (e.g. EPN, EPR, EPP, Ilfochrome 100 Pro, Agfachrome 100) shot under true daylight, they still varied in color rendition accuracy. So if you were shooting for a color-critical client (textile sales, or fashion industry) one carefully chose which film they wanted to use, and even made sure to use all from one particular production batch of that film.