I know there were machines made for the boat market where there is not the supply required to work a heater, and hot water is basic free as heated by cooling the engine.
They were expensive, as a niche market, mainly for narrow boats.
The basic idea is dirt like egg will change state when heated, so cleaning first with cold then heating up gets best of both. However last house dish washer went in space first used for H/C fill washer, so we used the hot feed.
It worked well, main advantage was it would cause hot water to flow, so hot water also at taps, but likely in real terms it was cold fill as so much water had to be drawn to get hot water to taps.
The same applies in this house, although DHW is heated with oil in Winter and solar in Summer so much water needs to be drawn, it would never actually fill with hot water, unless some water was drawn elsewhere. The other problem is hot water pressure is a lot lower.