When using something like CRI, they're evaluating the completeness of the spectral output...how much light is coming out at all the various wavelengths (colors).You can put a perfect filter on a questionable source, and NAIL the WB and G/M balance, but the light coming out of it will still look kind of "dull" or lifeless.
The problem is that there are no perfect filters for most LEDs. As these tests clearly demonstrate the spectral distribution of LEDs is so different from other sources (consisting of peaks and valleys) and different from one another that the use of color correction (CC) gels on LEDs can have unintended and undesirable consequences. Where you know what to expect when you put a color correction gel, say Full CTO, on an HMI or Tungsten light, you don’t know what you will get putting that same gel on an LED light. A good example of this is what happens when you try to convert the 5500K out-put of Phosphor White LEDs to 2900k with Full CTO gel. Where you can do it with some success with HMIs because there are long wavelengths in it’s continuous spectrum to pass disproportionately to the blue part of the spectrum to achieve a nominal 2900K, since LEDs don’t put out much beyond 625nm, there is not much for a filter to pass to rebalance the light output to 2900K, so the “corrected” light is too cool. Another undesirable consequence comes from the fact that Full CTO is designed to pass extra green (there is a bump in the spectral transmission curve of Full CTO in the green portion of the spectrum) and so it creates, given the amount of green inherent in Daylight LEDs to begin with, a disproportionate amount of green (creating an overall green bias) to the "corrected" light when used on Phosphor White LEDs (use this link for tests demonstrating this with a Lightpanel 1x1 Daylight Spot.) The gel pack that eventually made the Lightpanel 1x1 Daylight Spot in the test marginally similar to a tungsten light, was only able to do so at the expense of two stops – so much for the greater efficiency of LEDs.
Paul Dean