Contemporary Amperex Technology Co., Limited (CATL), the largest battery manufacturer in the world with a 38% share of the global market, has just announced some fairly significant breakthroughs in battery tech that aren't just theoretical; they're already hitting the market.
Why not use a large capacitor as a buffer. Would give peak power at the beginning of the charging cycle which is what you want anyway for quick turnaround.
Electricity is not my forte so correct me if I’m wrong but if you had a long gap in cars using the charger a capacitor would quickly become saturated. Which means you would need more wattage going to the capacitor at all times. Whereas with a battery if you have a car on a 5 minute charger with 5 minutes in between you could pretty easily run 500kW constantly to the battery and then as the battery dropped in level you could slowly ramp up the power draw