>water without minerals upsets the osmotic balance and leads to lysing (rupture) of the cells lining the stomach

I've heard many people claim this but I've never seen any convincing evidence that it actually happens in vivo. Even if you pour it directly down the throat, ultra-pure water stops being ultra-pure as soon as it hits the stomach. There's no way it can touch any gastrointestinal cells before it's contaminated.

Your linked article even acknowledges this. "currently available data have not unambiguously demonstrated a direct negative effect of low mineral content water on the gastrointestinal mucous membrane." It also cites an animal study showing no damage: https://www.ncbi.nlm.nih.gov/pubmed/8434446

Water can be an important source of dietary minerals, but the health impact of this depends on what else you're eating. I agree it makes sense to have minerals in drinking water, for the taste as well as the possible health benefits, but missing them is not actively dangerous.

It just gets more and more complex, doesn't it. Now we have to worry about remineralizing RO water.

    [Tap Water] 
         |
         V
    [Water Softener]------------+ 
         |                      |
         V                      V
    [Reverse Osmosis System]  [Showering & Washing] 
         |
         V
    [Remineralizer] 
         |
         V
    [RO storage tank]
         |
         V
    [Drinking]

In Len Beadell's book "Blast the Bush", he describes a solution (pun?) developed while building the Woomera Rocket Range. Apparently the groundwater had a terrible taste; so they built a distillation facility. But the distilled water was unpalatable because it was too pure. So, they experimented by mixing distilled water with small amounts of groundwater, eventually arriving at a drinkable ratio.

Yup. Potable vs. Palatable drinking water.

Yeah, I always wondered why there isn't simply one line running to the bathrooms + laundry room with softened water, and a separate line which goes through RO for everything else. Putting a RO + tank under the sink but the softener in the garage / basement seems backwards to me; I know nothing about plumbing but it would seem to make more sense to have a small softener under the bathroom sink just for the shower.

You put the softener at the inlet to the house because you don't want to have hard water flowing through the pipes in your house. Hard water precipitates minerals that ruin the piping, hence you put the softener at the main inlet.

The softened water is for more than just comfort of use though. It keeps the hard water from depositing in the pipes and any of the devices that have water pass through them. It was interesting when we installed a better softening system that our water heater started putting out hotter water since the elements got cleared of the deposits. It also stopped making creepy sounds.

From what I've gathered, hard water also lowers the life expectancy of the RO system.. Also you may want softer water for cleaning dishes, since softer water means better lather with less soap, saving your detergent.

That's actually exactly what I came up with when designing how I'd do it some years ago. I didn't bother as our tap water is good out here. The only difference is I had a greywater (or whatever they call it) system in the design for reusing stuff for garden, toilet flushing, water guns used on non-household people with minimal filtering, whatever.

Who's drinking demineralized water though? All of the bottled water commonly available for sale is not demineralized, at least to my knowledge.