Just read up on any existing quick-charging technology. Tesla's superchargers are the world's highest-capacity quick charging at 130kW, 2x the nearest competitor (CHAdeMO). And this still requires ~one hour to give the battery a full charge. Let's say you want to be able to charge in six minutes. That would mean something on the order of 1.3MW of electric power. So let's say you have a small-ish refueling station like a contemporary gas station, with eight charging spots. To power these, you need to pull ten megawatts from somewhere. The current power grid isn't dimensioned for this kind of demand, especially since the demand is very transient.
Tesla is attempting to solve this by building a massive battery storage capacity into the chargers and then trickle charging these batteries when the charging bays are not in use, so this is by no means a problem that has been solved once and for all.
Distribution level voltage is anywhere from 4kv to 25 kv. Say it's 12. Typical partridge or linnet distribution conductor might do 200 A. P=sqrt(3)IV cos theta = 1.73 * 200 * 12000 * 0.9 = 3.7 mw.
So you are correct your average distribution feeder probably does not have capacity for a 10 mw service. I guess that's not much of a shock.
In the United States the normal distribution voltage, prior to being stepped down for consumer use, 7.6kv phase to ground, 13.2 phase to phase. So yes, you're pretty close, except the 13.2 kv would be 3-phase. For heavy industrial use, 4.4 kv is common for actual end user equipment, but above that would be some pretty serious equipment for an end user. Larger transmission lines operate anywhere from 60kv on up to 1.1 million volts, so lots of power is possible, as long as you don't mind the brownouts.
Tesla is attempting to solve this by building a massive battery storage capacity into the chargers and then trickle charging these batteries when the charging bays are not in use, so this is by no means a problem that has been solved once and for all.