Why is Sake / Shochu used to make Mirin?

Finished Sake isn't used to make the mirin. But a scoby is. Like sake, mirin needs the bacteria in the scoby to make the sugars for the yeast.

So just adding yeast to the rice will not work. You need something for saccrification process (enzymes from bacteria or fungi) to convert the starch to sugar.

Edit: based on Pete and neels comments, if your koji-rice mixture has converted the starches to sugars then yes adding yeast will work by making a scoby.