I’m less certain about the other things - not many people will eg. put off a load of laundry to a less-convenient time to save 21 cents. And if my partner is too hot because the heater has been running for 30 minutes I really do not want to have a discussion with her about “thermal storage” :)
I believe in some places this is already done.
It is so much nicer to cool/heat pre-emptively. The power company has a very accurate view of what the next 12-24 hours will look like. If they know The sun will be shining bright and there will be an excess of solar energy, they should crank up people's AC so that they don't need to draw as much power later in the day. This way we smooth the demand curve without letting pipes freeze or giving people heat stroke.