Why does mixing inks make subractive colors?
The issue is that you're thinking about inks reflecting color, when in reality you should be thinking about the inks absorbing colors.

If the function of inks were to reflect colors, then by adding more and more ink (of a single color) you'd be able to make brighter and brighter colors, as more ink means more reflection. Instead, it's the white paper (rather than the ink) that's reflecting the light, and more and more ink (even of a single color) makes things darker because more and more light is being absorbed.

That's why mixing pigments/inks is a subtractive process - as you add inks, you increase the amount of light that's being absorbed.

So let's go to your example. Let's simplify things by considering only three wavelenghts of light: red, green and blue. Your red ink absorbs the green and blue light, whereas the green ink absorbs the red and blue light. Thus when you mix the two, each absorbs the light it would in isolation, and you have very small amount of blue light, and also very little red or green light, leaving you with a very dark, perhaps slightly yellowish-brown color.

Now consider subtractive inks. You have yellow ink, which absorbs just the blue light and you have magenta ink, which absorbs just the green light. You mix the two together, and the magenta ink absorbs the green light, the yellow ink absorbs the blue light, and the red light is free to be reflected, resulting in a red colored piece of paper from Y+M.

Generally speaking, modern inks (at least the standard, non-color changing variety) involve no chemical reactions in color production: mixing inks and pigments is solely a physical change based on absorption of light.