It's not exactly using the a*/b* CIELAB channels/values; even though that is what it looks like.
When you set the temp towards the blue side of the slider you are telling the program that the light was warmer (lower kelvin) and it needs to add blue light to the scene... and light is additive color (RGB).
However, anything in the scene that is showing a color is doing so through subtractive color (reflected CMY), and that has to be taken into account. So what you have is the CMY subtractive color (yellow) of reflected light showing in the image opposing the RGB additive color (blue) of the light source causing it.
Note that if you go to full additive on both temp and tint sliders (max blue and green) the result is a cyan reflected color; and if you go full subtractive on both (max yellow and magenta) the result is a red light source. And if you move them towards subtractive colors you are telling LR to display more light/color reflected, and the values will be lighter in a greyscale image (e.g. yellow/magenta are positive values in CIELAB).
Also note that the kelvin color scale is more likely to be familiar to a photographer and may be being used for convenience; much like how LR will display the RGB values in 8bit numbers even though it is working in 16bit (or 32bit for hdr). What the actual calculations are behind the results I cannot say; I am sure it is rather complex as visual perception is also complex and non-linear. E.g. if you add blue light to a scene it will affect something that is blue (reflects blue) much more than something that is orange (absorbs blue).
I can tell you that when you use the WB dropper in LR it is averaging the values for the selected point so that the RGB values are equal (greyscale); but it is doing that with a bias/offset based upon the surrounding points as determined by the sampling scale setting.
I believe auto WB looks for pixels where the RGB values are already nearly matched (i.e. nearly black/white/mid grey) and then applies the same kind of equalization of the RGB values with a similar bias for the other selected/evaluated pixels. e.g. it's going to attempt to avoid overcorrecting nearly black pixels at the expense of nearly white pixels (or mid grey pixels).