It's an interesting question. Us fish people usually know a bit about our local water quality. Here in Vancouver, our tap water is extremely pretty soft and even a little acidic (pH 6.8) coming out of the tap (freaking awesome for S American fishes). We have basically no measurable hardness or alkalinity, so it's very close to neutral pH with little to no mineral content, and in that regard, quite similar to distilled water, though no doubt contains other elements that distilled does not. My pet birds and I have been drinking the tap water for years (it's delicious) and the new puppy has been and will continue doing the same. I haven't heard of any warnings locally about adding minerals to the animal's diet or anything like that because of our soft water. So I would be inclined to think that offering distilled (or reverse osmosis) water would not be an issue for the majority of pets who otherwise have a well-balanced diet containing the correct proportions of the necessary minerals like Calcium, magnesium, etc...
I'm obviously not a vet though, and just a fish geek extrapolating out loud.
And RO water still has a fair mineral component, just much lower than the original. (but I doubt many people here have the ability to produce RO water
I think that for people who live in areas with high mineral content, bottled water (at least the bottled water I've tested) is similar to yours… quite soft, neutral pH. I think I'd still go with that rather than distilled water, at least unless I checked with the vet first. And regular bottled water is a lot cheaper than distilled.
And, honestly, while I know a lot of people who use bottled water for their Havs, I haven't heard many who were able to solve tear staining that way. I do use bottled water for Kodi when we're traveling, because we've had a couple of instances where the water changes on the road have caused him some tummy trouble, and poopy butt is NOT something pleasant to deal with on the road. Keeping him on bottled water seems to solve the problem while traveling. At home, though, he drinks our tap water, just like we do, just run through the carbon filter on our refrigerator. (Our water isn't as soft as yours, but still relatively soft… KH and GH @4.