[reducesuffering]

I've considered the "lost minerals" angle, and from my research, I don't think it's an issue at all. The main significant minerals in tap water are Calcium, Magnesium, and Sodium.

"For half of the tap water sources we examined, adults may fulfill between 8% and 16% of their Ca2+ DRI and between 6% and 31% of their Mg2+ DRI by drinking 2 liters per day." from https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1495189/

And another paper on calcium: "On an average basis this would represent 5% to 10% of the usual daily intake or approximately 6.5% of the adult RDA." from https://www.ncbi.nlm.nih.gov/books/NBK216589/

The calcium seems negligible. Magnesium is a top nutrient deficiency in Western diets, so could be somewhat negative, but because of that I (and probably you) should be supplementing magnesium anyway. Sodium is obviously found abundantly in western diets.

So based on that, I don't think re-mineralizing's tiny benefits are worth the potential risk of re-adding negative contaminants somehow.