Recharging would take a lot of energy but it's not unthinkable to have 1-3 GW supplied to an airport (which is what a reactor would likely supply), large metropolitan areas and large industrial factories already get to similar amounts. It's a couple of transmission lines and most large airports are close to population centers anyway.
The challenge would be getting the right amount at the right time, like now with quick charging. Like there, you'd probably have buffer storage at the airport so that it consumes electricity when available (e.g. during the day from solar) and dispenses it to aircrafts when needed. Luckily, most airports in the world have nearly all take offs and landings during the day, so there's a big overlap. Dubai would be an example where likely all would come from solar but a lot is needed over night (if we ever get electric long-haul flights).
So overall I don't think that this would be the limiting factor. But I guess larger airliners are more likely to run on synthetic fuels than electricity for a long time. And I guess that's fine, we have a lot of areas where cheap and/or dense batteries can help us much more in the short term (grid storage, cars, trucks).