...you still couldn't draw enough power from the grid to get enough energy to saturate the battery in that time.
Indeed, it's a substantial power draw required to charge a car in a minute or two. Gas pumps can put out about 10gal/min, so 2 minutes is a reasonable max time to use for comparison purposes. To get a Tesla's 85kWh in 2 minutes from 230-volt 2-phase, that would be about 400A, which is twice the power supply to a typical house. (And of course that 85kWh does not get you as far as the 20gal of gas would; but still I don't think consumers will be doing those calculations; what they'll be doing is driving away reasonably happy if the "fill up" takes a minute or two, but annoyed if it takes 10-15, and if it takes 30, that's a huge limit on acceptance of electric vehicles. Also, there's the economic model for the business, having cars drive up, spend a minute or two and pay, then drive away is a viable business. Having cars take 10-15 is not; too much land and too many stations required.)
So the "gas station" of the future, despite being an itty-bitty building, would probably need electric service similar to a smallish office building. Way more than they have now, but certainly doable, certainly power at a level that is pretty common for commercial developments. (And, BTW, they certainly wouldn't deliver 230V @ 400A, that would be a copper cable that would resemble current gas hoses in diameter, too heavy and stiff to be practical. I'd think they'd have to bring 2.5kV, or maybe even 25kV down to the building and out to the charging stations.)
But right now none of that matters, because batteries can't take charge fast enough. The current (haha) bottleneck in charging speed is batteries, if that limit is lifted significantly, then it would on to the next bottleneck. (Or, ideally, things get planned well enough to evolve roughly together...) And of course big batteries at the "gas station" have all sorts of potential uses as well: smoothing demand by loading up ahead of peak business, loading up at off-peak rates, storing the output of on-site solar, cheating and draining the batteries of cars when they hook up (OK, maybe that last one is not a sustainable business model)