@Anton Ivanov
The only particularly powerful 660cc engines in Kei cars are turbocharged. Trust me I've owned a couple of Miras (a 2WD and a 4WD) and while the engine can deliver 85bhp (more with a new chip) it uses fuel like a bigger engine. Sure my Miras were capable of 0-60 in under 9 seconds, but that was because they were light.
That's the thing about internal combustion engines, it's not really the capacity that governs efficiency or power output, but the amount of air you can make it pump. You mix the air with fuel and burn and the more air you use the more fuel you need. So in other words if you make a small engine move as much air as a bigger engine and you will use as much fuel. Using a turbo will make a small engine produce good power and or torque at mid to high revs, but it will do nothing at high revs, but it won't work at all at low revs and because you need a low compression ratio to deal with the boost you end up with an engine that is even more feeble at low revs.
The end result of which is that in order to get reasonable performance at lower speed you need to use plenty of revs, which will be less fuel efficient than a larger engine with a similar peak power output.
Austin Rover tried to deal with this by using a high compression ratio. This worked to an extent, but limited the boost they could use. And also meant they needed some clever (for the time) electronics to limit mid range torque in order to protect the transmission.
Lancia and VAG have tried using a supercharger for low revs and a turbocharger. This sounds like a great idea. Indeed VAG claim their engine has the economy of a 1.4 with the power of a 2 litre, if this is the case why do they still make 2 litre engines? Because if they put it in a big car it would prove to use just as much fuel as the 2 litre engine. The other problem is that such an engine is much more complex and expensive to make than a conventional engine and of course more expensive to service.
The reason for the popularity of the 660cc engine in Japan is down to the Kei car regulations. Basically you are limitted in some cities to 660cc, along with power, speed and size restrictions unless you have your very own reserved parking space. So the engines have evolved to meet a niche requirement. If they were the solution that you seem to think they are, do you not think that Japanese manufacturers would be using them in the cars they sell over here?
Yup, they're great for powering a Copen or a TR-XX, but they just wouldn't do the job in something bigger.