Don't you think the video's demonstration of colored LED's shifting hue/sat in the bell pepper examples as a predictor of white light of various spectra output isn't being cause by the light but through maybe software manipulation? I've never seen any of the white lights I've used do that to a bell pepper.
That site's info and video is made by very learned color scientists and engineers, so I'm surprised they attribute that color behavior to lighting. Most white lighting I've worked with applies a more flat filter effect as an evenly distributed wash/stain of the color temp hue over the entire scene but never to the extent they shift hue/sat that severely.
I've never seen a camera profile attempt to characterize that behavior and correct it applying the profile in post. Editing is further required.
Tim, what they are doing is rather unique. They are creating "white" light by selecting and changing the contributions of a large number of LEDs, each with a fairly narrow bandwidth. They have a diffuser so that you don't see any of the LED colors separately. And the "whites" are all computer controlled so that the chromaticity values are those of D60 above 6000K and black body below 4000K, avoiding the jump that is made at D50 to 4999K.
They then modify the spectrum but maintain the chromaticity of the white point.
Think of the specific example of your monitor. Let's assume you profiled it for D65 and filled it with white. That white does not have the same spectrum as D65. It just has the same chromaticity so it looks white and will appear to have the same white as D65 daylight. But spectrally it is VERY lumpy. One of the consequences of that is if you use your monitor's "white" output to illuminate say, a red apple. It will actually appear to be a more saturated red than if you illuminated it with D65 daylight at the same Lux.
What they are doing is demonstrating that by creating lumpy spectra that is still "white" one can increase the color saturation of many common objects. They believe this might be of interest commercially. For instance to enhance the appearance of produce or make product packaging stand out more.
And, there are ways to decrease saturation by moving the spectrum inwards from both ends a bit so long as they maintain the chromaticity white point. The whites still look the same but all the colors from the objects change.
Along the way they describe a couple metrics, one of which could be used as an improved CRI.