Watch any of the documentary series about life aboard an aircraft carrier [1] and you'll see that there's essentially no "just sitting around in the ocean." But your main point is still accurate: If there's power available, putting it to work making jet fuel might be a good trade-off for the additional drain on the nuclear fuel.
From friends and family in the navy: whenever you see a fleet or battle group "stationed" somewhere, it means they are basically doing right turns in the middle of the ocean (a joke being that someone has to balance the effect of all those left turning NASCAR races). While they are doing operations etc, a lot of this time is considered extremely boring, more so than even long voyages going somewhere. Further, they usually aren't going full speed, so to some extent they are "just sitting there", particularly in the effect that they aren't using the full output of their power systems - leaving lots of power and (if needed people time) available for fuel "creation".
Former Navy aircraft-carrier nuclear engineering officer here. The basic point --- that a carrier might be able to refine its own jet fuel --- is certainly valid. But so that future readers don't get the wrong idea: Any time a carrier is at sea, it does a lot of flight ops just about every day, whether for actual missions, for war games, or just refresher training. That in turn means most of the crew is hustling, usually 12 to 14 hours a day minimum, and sometimes 'round the clock. Maintenance has to be done in the gaps. When we were at sea, it was rare to get six hours of uninterrupted sleep in a night. Somehow I doubt that has changed.
