Sorry if this is a super dumb question but I'm baffled about why the rendered result changes with the settings

 Is it because color management just has nothing to do with rendering and it acts sort of like a ‘filter'?

Yes, that's a good way of seeing it.

Rendering turns a 3 dimensional Scene into a 2 dimensional image and the Color management comes after that and affects only the Image.

Yeah it's coding black magic. In your mind you had it as if color management is a post-effect you put on the image, but it's more like changing how Blender itself shows you what's on the screen and it just so happens that the image of the render is on the screen.