Perhaps installing a big battery bank that trickle charges would smooth the load.

How many MWs would a carpark roofed with solar panels generate? My calculations say you need about 4k square meters to get 1MW whilst the sun is shining bright. That's about the size of medium sized supermarket carpark. Double that if you cover the building as well. You might get a good 2MW at peak, enough to allow a few cars charging at once. Not much good at after the afternoon sun has gone down and the after work rush is happening at the supermarket.

Given most drivers will be charging at home, I wonder what percentage of cars we can expect to be wanting to charge whilst shopping?

Many DCFC car charges already use batteries so as to avoid more expensive service links while still providing rapid charging.

Is that true? If so, that's extremely interesting. However, for popular chargers, it seems unlikely to help pay the first charge of the day.

There is no "first charge of the day". The only first charge is the one following a power failure of the grid. During billing rollover, the charger's battery is powered up from yesterday's power.

That was a typo. Replace "pay" with "past".

Unless you have enormous battery banks, the first few charges will deplete them.