If you add all the minerals like you say and allow the water to evaporate, what do you think you will be left with? Could it be that the same rock formations will appear?
Calcium is one of the most important minerals...and this is what you will find in hard water, and no matter what form you add it back into the water when the water evaporates you will be left with solids. So, does it really make any sense to remove all minerals only to add them back in? Or does it make more sense to filter the water and remove most of the bad, while leaving behind some of the good?
Then, of course, there is the concern of the distiller's condensing pipes and what they are leaching into your water?
I tried distilling water and I found it time consuming, I didn't like the taste, adding minerals never tasted good either and having water jugs in the way was annoying as well.
My filtration system is easy and I don't have to play with the water, nor do I have to worry about anything leaching into it. It tastes great too!