While I'm understanding your head-canon approach, I have to say that the leap from the TOS aesthetic to the TMP one is the more logical approach, because you're going from a less-advanced design to a more-advanced design. DSC is basically showing the opposite of that.
Star Trek: Enterprise and the USS Kelvin in
ST09 already ripped that band-aid off though (unless you go with the "backwards ripple" explanation for the Kelvin's appearance).
The problem with this is the Discovery itself. Stated to be a brand-new ship, yet she has the older style nacelles.
Discovery may have been a brand new ship, packed full of science labs and spore drive tech, but the Crossfield-class could predate the Constitution-class. They just didn't see a point in updating the look.
That's my preferred explanation. I like to think there was an "original Crossfield class" contemporary with the other Starfleet ships introduced in
Discovery because...
*NCC-1030 and 1031 line up with the registries of the other
Disco starships, as does the name "USS Glenn" (I know the reasoning for the Discovery's # and yes, I wish the production people would stop doing this)
*interior spaces & exterior design of Discovery and Glenn jibe with the other ships introduced in the first episodes of the show
*Starfleet co-opted the mycelial network research "early in the war" and it's unlikely they could have designed and built a class of specialized starship to test the spore drive so quickly - Burnham comes aboard the Discovery a few months into the war, right?
*no "class ship" USS Crossfield was ever shown or mentioned
Let's say the Crossfield-class was designed as a carrier for fighters and/or landing craft and the initial production run was in the early 23rd century along with Walker, Cardenas, Magee etc. classes. Maybe the original configuration looked closer to Ralph McQuarrie's
Planet of the Titans concepts, or like the original design of the Discovery from the first teaser (which I prefer over the final look).
Perhaps the need for fighters and landing craft never emerged or advances in technology rendered them useless (improved transporters?); anyways most the ships get scrapped or mothballed by the 2250s.
When hostilities with the Klingons commence in 2256, Starfleet scoops up Stamets' mushroom research and needs testbed starships for the spore drive. They refit two mothballed Crossfield spaceframes due to the ships having large internal compartments originally for shuttle/fighter/landing pod storage and maintenance - this becomes the "systems hub" and the spore cultivation areas we see on the USS Discovery.
*There's no scratches or wear on the Disco and Glenn because the ships never saw action OR were extensively refitted OR were spaceframes that were never finished due to the retirement/cancellation of the Crossfield class
*IF you want the two Crossfields we see in Discovery to be newbuilds, you could say they were assigned NCC numbers that went previously unused for whatever reason (this is my explanation for out-of-place Constituion class registries on Constellation, Cayuga, Intrepid etc.)
*the McQuarrie-style starships we glimpse in the background of productions set later (spacedock in
Star Trek III & Wolf 359 wreckage/Qualor II surplus depot hulk in
TNG) may be other classes of carrier or large transports