How to correctly make the texture black and white for making it colorable

In game I'm giving random colors for humans. Like clothes, hairs and skin.
I tested how coloring works with hairs and it is perfect:

(How does it look like in photoshop)


(Applying color in photoshop with "Multiply" modificator)


(In-game/Unity view with same color like in photoshop)

But when I tried to make colorable the skin, all became bad, because I can't find needed black&white colors to make skin look good:

(Photoshop view, used same colors of skin for left(static) image and right(black&white) image)


(And how black&white looks like)

So, what I'm doing wrong to make skin colorable?

 

what you want to do cannot be done that way

black and white cannot account for saturation, therefore changing 1 color slider will never give you the results you want

for this effect you need to use a shader that changes colors, the easiest you can check is shader graph tutorials for color swap

 

Okay, thank you!