I would assume that if you took some shots with your camera (lens covered with the ExpoDisc) of an incandescent source and bracket the shots with manual settings of CCT, as you approached the actual CCT of the light source, the recorded image would approach "white" ? Yes?
If we were talking about absolutes from the flashlight, yes it would be white. But in the context of computer displays, there is an extra complication. In nature, when more light is added, it just gets brighter. In a computer, white is already max, so the only way to show bright (accurately) is for everything else to get darker. So adding color reduces the brightness of a given sample. To be on the same K scale as the colors I'm showing then, white has to be darkened to gray.
Here's an example Mag85 (incan) with a measured color temp of 3500K:
The left half is what the camera sees, with 21 different K settings, starting with 2500K and ending with 4500K (zero plus or minus 10 whole steps). Down the center is a pure gray reference. Side note: you may notice that at 3500, the actual color is still a tiny bit blue. But 3600K is just a tiny bit red. Even 3550K looks a bit off, the true color the light is putting out is between the two values, but K (expressed in 00's) is not precise enough, so we accept it.
The right half is what happens when the same values are adjusted so that 3500K is near white. But as you can see, as the brightness reaches the computers ability to display pure white, the colors are almost pure white as well (as we approach maximum brightness, colors get washed out and look rather :sick2
.
So to answer your question, as the CCT approaches the value of the light being measured, the image appears gray.
Or does the image approach the spectrum of color one sees at that color temperature?
The spectrum is there, but as the goal of this approach is to show the tint, the tint is all you see. With a camera, the only way to show the spectrum of color contained in a light source is with a prism or by shining the light source on something colorful and showing which colors render/reflect.
Does the manual adjustment of the camera attempt to filter out the tint and give you a white appearing white surface when you have set the color temp correctly for your light source or does the camera attempt to give you an image of a white surface tinted to match the spectrum of a black body, at that kelvin level?
From the point of view of the camera sensor, they are the same thing. Cameras recording light are much simpler than what you have to deal with, coordinating the output of a given LED with the perception of the human eye. Since all the camera needs to do is calibrate to achieve white/gray, a black body radiating at 6000K blue is just as easy to coordinate with as an LED radiating at 6000K green. Configuring the camera to match 6000K will neutralize both tints with equal ease.
Part of the confusion here is that I have introduced white balance disks at the same time as a new technique that mis-uses them. Normally, they are used to erase tint. Figure out what the tint is and calibrate to it, the camera's K moves about to adapt. Here, I have frozen the K setting of the camera, so the tint is what your actually viewing.
Think of an ExpoDisc like a grey card, except instead of measuring the reflection of light from a given surface, you are measuring light on what would be, its way toward the grey card. As with a grey card, the goal with using a Disc when taking a photo of something is to match the CCT of the camera to the CCT of the ambient light, so the final photograph appears as though the ambient light was white.
Ironically, most of this came about because of photographic improvements. Back in the dark ages, we used film. Film was calibrated to sunlight and we used it outside all over the place in natural light. It wasn't sensitive enough to take indoor shots without a flash, so indoor shots were calibrated to a single known source, the flash. And even when color was messed up, someone else was doing the developing and often correcting the color for us. Then along comes digital. Suddenly we are doing everything ourselves, but more importantly, we now have the ability to take indoor shots with ambient light. And as it turns out, the CCT of indoor lighting is all over the place,
creating this problem. But I can't stand flash photography, so to me, this is a good problem.
BTW, here's a good primer on photographic white balancing:
http://www.ronbigelow.com/articles/white/white_balance.htm