Yes reading up on color zones this is exactly the functionality. Coming from color grading video-material in Davinci Resolve, this is one of the most powerful tools there is - not usually found in photo editing software.
I don’t think Lab is a prerequisite to make it work. In Davinci Resolve, the 9 curve combinations are usually done in Davinci Wide Gamut or Aces. Even works in Rec. 709 although more prone to artifacts with the limited color range.
Yes it is possible to have all three curves affecting the same value - this is what makes it so powerful.
E.g. using C/C to desaturate only the oversaturated colors. – Then using C/L to desaturate the highlights – Then using C/h to boost greens and reds, making skintones and foliage pop.
Using h/h and h/L would also make it easier to achieve things like the dark blueish green tones that Nosle was describing in the Spectral film simulation thread: