Why don't we think about battery storage for all sources of electricity on a utility scale instead of just renewables?

Taking, at random, Overland Park, Kansas, as an example:

• Population 191,278 (2017).
• Area 195 km².
• Annual energy demand (per capita) 13,500 kWh = 37 kWh/day. World Bank.
• City demand = 191278 x 37 = 7 x 10⁶ kWh/day = 7 x 10⁹ Wh/day = 3600 x 7 x 10⁹ = 25.5 TJ/day.

For pumped storage the formula for energy stored is \$ E = mg\Delta h \$. Assuming we could create a pair of lakes with a Δh of 100 m somewhere nearby then we would need to move \$ m = \frac{E}{m \Delta h} = \frac {25.5T}{9.81 \times 100} = 25.5 \$ million tonnes of water to the upper lake to store one day's worth of energy. That's 25.5 Mm³ in volume.

Making a lake the size of Overland Park we would fill it to a depth of \$ \frac {25.5M}{195 \times 1000 \times 1000} = 130 \ \text m \$ which is deeper than the 100 m we suggested raising the lake to.

The point is that the energy requirements are huge and any storage system would have to be equally huge. You can find battery energy densities on Wikipedia.

Last time I looked online battery storage was a little below US$200/kWh. That requires an investment of 37 x $200 $7400 just for you and $1,415,457,200 for your city for a one-day battery backup.

I am not sure exactly what you are expecting as an answer, but storage has already started to be used to supplement all energy sources. Utility-connected battery banks have already proved superior to any alternative peaker plant.

This report on the Australian 129MWh Tesla battery installation's first year of operation could shed some light on it. In just one year it has pretty much recouped more than 75% of its costs (33% to the owner and the rest in savings due to its reducing FCAS market pricing by nearly 90%).

Conclusions from the report include that the battery system has contributed to the withdraw of the requirement for a 35 MW local Frequency Control Ancillary Service (FCAS), decreased the South Australian regulation FCAS price by 75%, helped connect South Australia to the National Electricity Market, among various other contributions.

That means that 35MW of fossil-fuel-powered peaker plants can be removed from the market. Given its near instantaneous response to demand, it also means that utility energy markets might start pricing speed of response differently, which will further benefit these types of storage.

Just adding enough storage capacity to sustain the grid while a fossil-fuel plant is being wound-up, is enough to provide some CO2 savings. Extrapolating from the Australia installation, savings could be at the very least on the order of 30% of the installed battery generating capacity.

Replacing peaker plants is an explicit market focus for Tesla, which as of this writing has installed more than 1GWh of utility-connected battery banks worldwide.

Part of your basic premise is false: gas turbines can go from zero to full power in a matter of minutes. They've been used to provide peaking power for decades, since it's rare for power demand to vary so rapidly that they can't handle it. Because traditional generation capacity can be changed so quickly, there's no need to store electricity on more than a trivial scale.

The reason we're looking at battery storage (and flywheel storage, and pumped-hydropower storage, and a whole lot of other things) for renewables is that they can't be used to generate power on demand. If the grid operator sees that the Superbowl halftime show is coming up, they can instruct a gas turbine or two to start up to deal with everyone microwaving their snacks at the same time. But they can't turn the Sun on at night, or order the winds to blow harder. Hence the need for large-scale storage of renewable power.