My fairly new gas hot water heater is located in the garage where I brew. I put a tee on the output side with a valve so I can draw hot water right from the tank as needed. The water is taken right out of the tank and does not travel through the house plumbing. There is no detectable difference in the taste of cold or hot tap water at my house.
The morning of brew day I crank the thermostat up to max and can get 150 deg water. That water is heated a bit more on the burner for mashing, but it does not take long.
I used to use that water (via a white RV hose) to supply the HLT which reduced heating time there as well. After a few experiments, we now have eliminated the HLT entirely and use the 150 deg water directly for sparging. No measurable loss of efficiency or change in wort collection was noticed so it has become our new SOP. After sparging, the thermostat is set back to the everyday setting.
My big system ends up using about 80 gallons of hot water on brew day (2 barrel batch size) so all this activity has to be coordinated with others in the house who might need to use the domestic hot water for showers.