They easily do. Discharge rates are typically higher than charge rates. For stationary batteries it's all easier due to being able to have larger, more parallel batteries, and better cooling when weight is not a concern.

Battery-backed charging stations are already common, because it allows use of cheaper grid interconnection, and use of cheaper off-peak or renewable energy.

If the car batteries can handle some amount of power, batteries on the other side can handle the same amount.

Especially because the station would want to have multiple cars worth of energy stored, which means the load is divided among more cells and they don't have to work nearly as hard.