You would need an impractically sized cable to carry the voltage required to "fast charge" a plane battery, and that would still be much slower than using a liquid fuel.
Wrong. With planes, it'd be trivial to just swap out batteries at the airport. Aircraft aren't like cars owned by individuals. With aircraft, they're owned by large companies like Southwest or Delta, who have whole wings or even concourses, plus giant hangars, at every airport they have a presence at. If they actually had batteries sufficient for this task, they'd simply swap them out at each stop, just like they refuel and swap out the baggage right now.
They're already talking about doing exactly this with cars, but with cars there's some problems with the idea. Battery packs aren't fungible like fuel; an old one doesn't work as well as a new one, they can develop problems and need repairs (like replacing individual cells), so if you swap out your brand-new $20K pack for a worn-out old one, you're in effect losing a lot of money. So they're coming up with ideas like leasing the pack when you get a swapped one, and later returning for your charged-up old pack which you own, but that's obviously problematic.
All this goes away with aircraft, since the airline company and the company swapping packs at the airport are one in the same. When a Delta plane lands and docks at the jetway, Delta personnel will swap out the pack for another Delta-owned pack. Delta will manage all its own battery packs, so issues of who owns which pack and who's responsible if one develops problems all go away.
Obviously, this discussion is academic since we certainly don't have batteries capable of storing the amount of energy needed for an electric airplane, but if some giant breakthrough in battery technology is made next year and all of a sudden, electric airplanes are totally feasible and completely capable of replacing modern jet-fuel-powered aircraft, the recharge time will NOT be a problem; they'll just swap batteries the way people do now for rechargeable power tools.