Cost is not the only variable. Efficiency and charging/discharging capacity and time are also important metrics. Basically storing heat and converting it back

Cost is not the only variable. Efficiency and charging/discharging capacity and time are also important metrics. Basically storing heat and converting it back to electricity, you are going to lose quite a bit of energy. And that's before you consider cooling effects. Heating up a large mass is not particularly fast. It takes some time. And then getting a lot of energy out rapidly is also not really that easy. A lithium ion battery has probably well over 90-95% efficiency. It keeps its charge. And you can charge/discharge rapidly. Basically, you cans switch them on fast enough that switching from the grid to a battery when there's an outage, you probably won't notice the switch over. Both have their strong and weak points and there is probably going to be a use for both on the grid.