It's not so much that it's a race of course - the rapid response was always needed, but the comparison is between the battery and having a gas plant sitting there fuelled and turning over but not actually generating. The battery can do that more cheaply.
That's what I (and various other commentators) used to think. However, the way it's playing out seems to be slightly different.
In the early days of supercharging (and other fast charging) it seemed like a good idea to have batteries so that you could install a small site without needing to add a new grid connection - trickle charge the batteries from whatever supply already exists on site (maybe even solar) and unload them when a car turns up. Evtronic sell a 50kW CHAdeMO/CCS unit with batteries inside. This makes sense if the level of demand is only enough to require one or two stalls - a site like that can only have a very low capacity factor (if users are to be satisfied with the quality of service) and so can spend most of its time recharging at a slow rate. However, Tesla are already well past that point - once you have enough demand to need 8 or more stalls, then the capacity factor is higher and you need a largeish grid connection anyhow, even if you have batteries.
So we are now talking about batteries to manage peak-hour energy costs. Fundamentally, charging at peak hours is going to be expensive: you can either buy electricity at expensive rates in the market, or you can own your own (expensive) battery to let you time-shift the power. However, owning a battery and just using it to offset charging is a bit wasteful: you make more money using the battery more generally to attack peaks in demand regardless of who caused them rather than limiting it to strictly your own local use.
When you start thinking of it in that way, it no longer particularly matters where your battery is located. If you have plenty of spare space at the charging site, then putting the battery there is good economics as a single grid connection can serve both of them (without even needing to be any bigger). This is why we are seeing batteries being co-located with solar farms and wind farms (and have also in recent years seen diesel generators co-located with solar farms, as a cheap way of providing 'generation resource of last resort') without needing a separate grid connection.
It is notable from that planning application I linked above that it's not actually Tesla's battery. They are manufacturing it, but it's actually being installed on behalf of Camborne Energy Storage Ltd, an investor in energy projects.
