That's an interesting point. Theoretically though any fluid with a lesser concentration of minerals and nutrients will flush some of these out of your body if I understand diffusion correctly.
You could add some healthy minerals back in by adding a small amount of sea salt (NOT table salt) to your water after purification.
Anyways, rainwater is essentially distilled and then condensed around a dust or smoke particulate. It's probably one of the safer water sources. But I see what you're saying about maybe wanting to balance the pH and mineral content first.
Yes, but few people drink rainwater. Even if you do it hits your roof and then collects in pots.
Most of us for the last six thousand years drink spring water or water from streams or wells.
I used to teach Chemistry and the fact that water is "neutral" is quite misleading. It is both basic and acidic, which means it can really dissolve anything. If you drink well water or spring water you take the edge off the water because it has already had a chance to dissolve stuff from the ground. Most of what water dissolves from the ground will just pass through you. Some exceptions would be calcium and salt.
No one would drink Sulfuric acid but why? Sulfuric acid has free H ions floating around, but so does water. The difference is quantity and the fact that water also has free OH ions, like lye.
Drinking one glass of distilled water is fine, might even be good for you. But drinking it exclusively day in and day out, I don't think that is a good idea.