The numbers for that chart are derived from a machine (spectrophotometer) measuring the spectral response of each color, gray and black patches under a given amount of light and color temperature illuminant as Eric indicated. What that spectro "saw" is by the numbers and doesn't take into account human's adaptive nature to brightness, contrast and saturation of colors to any given scene viewed.
If you want the scene as seen by the spectrophotometer to correlate with the photographic image, you will have to adjust the viewing conditions such that the adaption of the human visual system is the same for viewing the image as for the original scene. I downloaded your corrected image and looked at it with Imatest Colorcheck. The DeltaEs are small, indicating that your profile did quite a good job or reproducing the chart. When comparing your image on my calibrated monitor to my own ColorChecker, I see a good match even though my monitor is 6500K and my Solux viewing lamp is 4700K. My visual system can make the necessary accommodation even for this mismatch of color temperature.
This is much like the affects on perception when the lights are turned on in a darkened movie theatre where our eyes immediately see a loss in contrast and richness. A spectro if it were possible to measure from the movie screen would still see the CCchart color patches as being the same because the spectro's is using its own light source and not the movie theatre's.
I don't think that is true. To measure the screen with a spectrophotometer, you would have to have an instrument that reads the actual luminance of the screen, not a reflection instrument that uses its own light source. In other words, you would need a radiance pixmap of the scene. When you turn on the theater lights, you not only change color and brightness adaption of the viewer, but the theater lights dilute the luminances on the screen, and this is most prominent in the shadows, since the luminance of the theater lights is added to the values produced by the projector, and the effect is most marked in the shadows, where the luminance produced by the theater lights is much greater than the shadow luminances produced by the projector. The effect is analogous to flare light, which washes out the shadows more than the highlights. If you wanted the image to appear good with the lights on, you would have to greatly increase the luminance of the projector so that the projected shadow luminances would not be overwhelmed by the theater lights.
If I am looking at a reflection print and increase the ambient illumination, the highlight and shadow values are increased linearly according to the change in illumination.
The image samples below I took with my Pentax K100D DSLR in an attempt to get the gray and black patch readouts to measure as close as possible to the published Lab numbers using a curve adjust. My illuminant and light source is direct sunlight for all shots. You can measure yourself how close I got. The last image shows the profile applied along with the settings that gave exact Lab numbers to a real scene captured under the same light intensity. Note the lack of contrast of the overall appearance of the surrounding scene. This is the same effect that happens when the lights are turned on in a darkened theatre.
I can tell you for sure my eyes saw that scene as much brighter and full of contrast than what's depicted.
The color checker is a low contrast scene and the luminances are easily within the range of the monitor and even a print, so the scene can be rendered without any luminance compression. However, an outdoor scene has much greater contrast and a linear rendering looks flat, and a sigmoid tone curve can help fit the luminance of the scene to that of the output medium. For a HDR scene, a global tone curve fails and local adjustments are necessary. This is the difference between scene rendering and output rendering. If our monitors could reproduce the actual luminances in the scene, no luminance compression would be necessary and we could use the scene referred image directly.