Your analogy is flawed: let me fix it for you. "I invented a battery powered coffee machine which works using 10000 times less power: it works by deferring the actual coffee making to another machine you plug into a wall socket. Now you can stop worrying about recharging the battery on the first coffee machine, and still enjoy your coffee just like before!"

With WiFi, the advantage is that battery powered devices like your smartphone will consume less energy. Sure, your router will draw more power from the wall socket, but you are effectively reducing power consumption where you actually are using WiFi the most, mobile devices that run on batteries.