My skepticism is heightened by the fact that good old Lead-Acid is crazy cheap compared to the classy stuff light enough for cars, (plus expertise in the care and feeding of large battery banks isn't exactly hard to come by in telco, datacenter, and solar-power sectors), and I've never heard of anybody using those for peak/off-peak optimization, even if they have them anyway for backup during power cuts.
I might blame mere stodgy conservatism, except that on-peak/off-peak and capacity optimization in heating and cooling systems (eg. small chiller runs all night, gradually freezing a big brine tank in the basement, chilled brine is tapped for cooling all day instead of having a big chiller capable of keeping up with solar heating and occupant/hardware generated heat running full bore during work hours, various schemes for absorbing, storing, and slowly re-radiating solar heat in colder locations) is something that has been explored, and not just in fancy uneconomic tech-demos, in newish buildings. Retrofitting the old, pre-oil-shock building stock isn't always worth it; but the numbers often add up for new builds. If the same thing could be done for power, I would have assumed that somebody would have tried it.