Yeah, this should really be the future. There's a lot of unnecessary materials used/energy wasted to give us our current "all power costs the same all the time" system.
According to this, about 70% of US household energy use is heating/cooling the space, or water. Much of that can be time shifted. What can't be time shifted can be stored in cheaper ways than battery storage.
1 tonne of rock heated (or cooled) 20° C above ambient is a store of about 4.7 kWh. According to that same site, the average yearly energy use in the US is 10500 kWh. If 70% is heating/cooling, that's about 20 kWh per day, so you'd need about 5 tonnes of rock to hold that enough energy. That seems like a lot, but it's just about 2 cubic meters of rock.
If you use water, it has 5 times the specific heat (but less density), so you only need 1 cubic meter. Probably easier to heat/cool/use, too. Water can also be heated more than 20 degrees above ambient, too.
Really, we should create incentives for homes to be built with high thermal mass. Even without any sort of fancy direct heating or cooling of a thermal mass, it will store significant heat.