^That's a common enough school of thought, however it overlooks the somewhat more common school that likes to think of NCC numbers as linear, thus placing all the FC designs before the launch of the Galaxy class in the late 50s (at the latest). Thus, many people just think that "design cues" are simply from a different group of designs that have stayed similar through several decades. Inasmuch as some case designs (Volvo, VW, etc.) stayed the same for many years concurrently, and how US or Russian warships can be distinguished by certain design cues regardless of what decade you're looking at.
The things is, we can't just write off all registries being linear, as there's evidence that they are not chronological. For example, the 5XXXX Oberth class Tsiolkovsky's dedication plaque stated that the ship was launched only a year before the Enterprise-D. Plus, the Pegasus was the same type of ship with another 5XXXX registry, and it was stated in dialogue
that it was only five years older than the Enterprise-D. The brand-new Prometheus class prototype had a 5XXXX registry as well.
Plus, we have the Ambassador class, which visually is far more advanced than the older Excelsior class, and yet the known ships have registries of only 2XXXX while the majority of Excelsiors we saw have regs of 4XXXX (not to mention the even older Miranda class, with regs of 3XXXX!)
What this has to do with the OP is that 1.) I think the "Galaxy family" of ship classes (i.e. the New Orleans, Springfield, Challenger, Cheyenne, Freedom, and Nebula) were all built around the same time (circa 2350-60) even though they mostly have 5XXXX and 6XXXX registries; and 2.) that the FC ships are actually new, post-BoBW ships regardless of their low registries. I understand that two ship classes with drastically different design lineages can exist at the same time (i.e. the Constitution and Oberth classes), but there are more similarities to the Sovereign than anything else (the bridge modules, the escape pods, the angular style of the nacelles and saucers.)
Actually I tend to go with the far easier "ILM didn't give a hoot about the registries" route.
The FC designs were only intended to be background ships (as clearly evidenced by the low quality nature of the models in terms of both textures and polys - the slight exception being the Akira*), so I imagine they gave very little thought to it.
Furthermore, several of the design cues present are similar to Defiant
(see the Norway
-class) which I'd think we should all be able to agree was pretty much a first among Starfleet designs.
Of course, there's always the possibility that the background ships looked substantially
different when they were first commissioned, and that the styles seen in FC are indicative of heavy refits and modifications - likely in the wake of Wolf-359 itself (it's not as if we don't have precedent for it, what with the Constitution refit... and I also adhere to that rule for the low-registry Grissom seen in TSFS)
* The Akira I regard almost certainly as having had a partial refit, what with the saucer, registry, clunky nacelles, old-fashioned deflector.... and then the new-fangled lifeboat hatches slapped on top.