What to make of the looks of the ship?
We see artwork in which the ship labeled CVN-65 retains the square SCANFAR radar arrays around the island (Archer's ready room), and artwork of a nameless, registry-less ship with SCANFAR (Picard's lounge), both without a timestamp. We also see up close a ship labeled CVN-65 without SCANFAR, with a Forrestal-style bridge instead (ST4:TVH), in the eighties.
In our reality, USN never wanted to cough up the dough for removing the SCANFAR supports even though the old radar itself was long gone. In the Trek reality, they may have. Or then their CVN-65 was completed without the structures, but sometime post-1980s the USN added supermodern electronically scanning radar systems to the island, incidentally looking a lot like our old SCANFAR.
We might speculate that our future artwork depicts the flat-panel island version of the ship because it was in that configuration that she performed the heroic feats for which she is remembered (and indeed considered even more worthy of immortalizing in decorative art than the old CV-6, which did all right in WWII but only ever got that one appearance in TMP). And we can thus choose between pre-1980s and post-1980s.
Might be it's easier to outdo CV-6 post-1980s, if there's war like never before. And the one thing we can rule out is there having been an additional World War before the 1980s but after WWII and CV-6, because the running number is only up to III when Man finally stops fighting Man.
OTOH and IMHO it would be a good idea to extend the service life of CVN-65 to the 21st century, so that she can assuredly take part in WWIII, even if we decide the Eugenics War/s was/were a completely separate thing. All the more glory to the ship, then; a longer stretch of her life spent in that cube-island configuration; and an opportunity for more interesting death scenes involving futuristic weapons.
Timo Saloniemi