"...I am willing to guess that many (not all) people who use distilled do not have soft water. ..."
No, that's not what distilled water is all about. Soft water, hard water, filtered water, ozone treated water, spring water, city water, well water, tap water, etc. ... all have dissolved minerals and metals in them. If you are having good luck making soap using water from the tap, then by all means keep using it.
Distilled (or lab quality reverse osmosis) water do not have these contaminants, so if you're running into problems with spots of DOS cropping up in your soaps, you may want to switch to distilled to see if your water is the culprit. DOS can be triggered by calcium, copper, and other dissolved materials in non-distilled water. These materials will NOT be removed except by distillation or laboratory quality RO treatment.