Didn't we used to pump steam around for heating and now homes have individual furnaces? If the cost for this were about the cost of a furnace I don't see why this wouldn't be viable.
That's a great example of why this is a bad idea. District heating is a lot more efficient than using individual heat sources. Especially so when the heat being used is heat that would otherwise go to waste (e.g. waste heat from industrial processes).
I think we can have both? I don't think anyone is advocating for one giant battery. In my ideal imaginary world, there are medium sized battery parks near every transformer station. In that same fantasy, the foundations of wind turbines are also batteries, so I shouldn't pretend that this is hard science.
First-gen district heating was steam, then moved to hot water. It didn't go away [1] but there's regional differences. In cold climates there are oil based simpler backup plants that can cover for the normal CHP[2] and elctric+heat pump based generation. It's a good way to buffer and use cheap energy in periods of cheap electricty from eg high wind production[3].
In Polish cities there are attempts to move in exact opposite way (close individual furnaces) due to rampant pollution during winter (1000% of PM 2.5 norm level of pollution happening occasionally, over 400% PM2.5 norm lasting for weeks etc)
I bought a generator for just this situation.