Earlier quoted context omitted.
I'm not convinced. Picking one fresh-looking color, and sliding the "H" slider, I'm running into non-fresh (dull) looking colors. Or is that somehow supposed to happen?
In a sense, yes. The true "freshness" of a color is shown on the color picker as the distance from the center of the circle. This is equivalent to the chroma component of CIELUV, the color space that HSLuv is based on. The problem with that component is that the maximum chroma is different for different hues, so if you were to move the hue slider while keeping true chroma static, you would run into impossible colors…
Yes, that would be nice. But as I understand it, fixing C would cause the H slider to show gaps too (?) Also, if you show sliders for H, S, L and C, then I think it would not be clear from the interface which sliders are fixed.