On the contrary, it is limited by the size of the cable going from the electricity distribution grid to the side of the car. With a typical (not a high performance) car engine capable of generating 75kW of mechanical power or so and a typical wall socket (in the UK at least) capable of transferring 3kW (and the average house having a 24kW total limit), there's going to be a problem that can only be solved by upgrading the cabling and using special high power sockets.

That's with a 120 kW charger. Can they increase the power 6 times? 720 kW per car?

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.