Martin, we simply do not have the infrastructure to fast charge a whole load of battery powered electric cars. To charge a battery powered car as fast as you currently fill a fuel tank would require a massive current.
The increased load on the grid would be potentially enormous. How many cars accross the country are currently being filled at any given time? Multiply that by the current draw of a single car being charged in five minutes. Then consider how much we would need to spend upgrading power distribution to cope with it. There is little doubt that every filling, sorry charging, station would need it's own substation, possibly quite a substantial one. Then of course they would need the feed off the grid for that substation.
All of that doesn't even address the issue of the cable needed to plug the car in. You probably wouldn't be able to lift it. Then of course there is the cabling from the charging point to the batteries. It would probably have to be much thicker than it is now to carry the extra current. Then the batteries might need to be chunkier too. So you would be adding weight, which would of course reduce the range and increase wear and tear on the running gear.
The whole argument around any alternative fuel source for vehicles is massively complex, being a morass of relative pros and cons. Chucking around simplistic glib statements helps nobody.
Politicians like things to be simplistic, hence their wholesale adoption of virtually any alternative, be it the apparently now out of favour bio fuels, battery cars or their current darling the hydrogen fuel cells. All of them introduce as many problems as they solve, most of which centre around the supporting infrastructure. Any fool could have forseen the problems caused by biofuels, but the politicians and money men missed them. You'd expect the tefal heads to miss them, because they are very focused and can't see the big picture.
One thing that is important is that a conclusion is reached on the way forward. It would be incredibly wasteful to start on a number of different infrastructures only for one to be finally proved the "best", thus rendering the others obsolete. Betamax may well have been technically better than VHS, but how much Betamax gear was replaced by VHS? The waste of money and resources are something the world economy and envirnment could well do without.
Another issue is that of taxation. Supporters of every alternative may trumpet it's cheapness as a major plus point, but we must bear in mind the loss of tax revenue should petrol and diesel be supplanted. What will governments do to make up the shortfall? They will of course tax the alternatives. Politicians may like portray high fuel taxes as "green" taxes, but they are simply taxes.