Recolor artwork "Color wheel" is distorted?

You wrote: "How did you get to that thing you are showing there?
My reply: The document you viewed is an AI document saved as PDF. Please open it in Illustrator to see. That should answer this question.

You wrtote: "The RGB color wheel  in the Recolor Artwork is based on the Natural Color wheel"
My reply: Do you have any Link, perhaps, to this "unique" Color wheel? I'm aware of several Color wheels but never came across anything resembling "Natural". 
You wrote: "The other thing is color management settings that might shift your color points"
My reply: No. The geometrical (hue angles) representation of color is purely abstact. Try changing the document color profile from sRGB to WhateverRGB to see for yourself: the absolute position of each color on the wheel does not change at all. Not that I can see. Please take a look at the two Photoshop PNG documents: both are screen capture, one show the dialog box while the document profile is ProPhoto space (huge space) while the other is sRGB-based.

For comparison, here is the Color wheel panel in Photoshop :

Do you see the way the colors are organized around the circle? See how Yellow takes a tiny, wheeny part of the circle? Compared to how yellow is "expanded" in Recolor Artwork?

Sadly, there is no Color wheel in InDesign. 
Finally, here is what I would consider the most "Natural Color wheel", based on RGB geometric properties, how it is represented in color science, shown her in Matlab: on the left is an RGB image and on the right the projections on the HSB space, showing only colors whose Brightness values are comprised between 75 to 100. You see where the dots cluster? And notice the location of each color on the wheel, it is the same as that used in Photoshop: