Not_a_boffin
I think you'll find the only FJ ac we have in the pipeline is the F35C.
The trouble with tying your hands to such a short runway (even with a cat 'n' trap) is that you'll still be severely limited with what weapons you can get airborne (and therefore what effect you can deliver), especially in +50 deg C. On a 13kft runway, this isn't an issue. But perhaps F35 is so good that it will be able to carry 2 x Stormshadow or RAPTOR (or equivalent), I really don't know. My major experiences of STOVL ops is that they suffer hugely from what they can deliver because of weight limits, both on take-off and recovery.
And my other 2 major experiences aboard US carriers (Sarratoga & Abe Lincoln) were that although the ac (A6 and then F18) were highly capable, the mission briefings spent more time discussing departure and recovery than anything else. Indeed target discussion took less than 10 mins; stark contrast to my land-based experiences where departure and recovery are "SOP" and everything is focussed upon the targeting. Perhaps things have changed.
So, yes, a carrier gives you extra freedoms/flexibility that may not be available to land-based fast air, but it also limits the effects that can be delivered - and that's what we're they're for after all, however we decide to go about doing it.