Yup. And that "big battery" will be charging one car per day so it could recharge to provide that megawatt fast charge.
I still remember working at the oil field 40 years ago where we had three 12MW compressors pumping gas into oil wells ("gas lift"). It is like charging 36 those fast charge cars. Nothing special, just a little thing -- we had a couple of dedicated high voltage power lines straight from the power plant...