We don't know whether there are refits still out there. The Federation is massive, there could still be whole armies of refits out there. We saw a secondary hull in the aftermath of Wolf 359.
Indeed, if the
Constitution-III was a relatively new class of ship, and if (based on precedent) the prototype was a rebuild of the
Constitution-II (which is probably bigger than the current official size thanks to SNW, so it might not be that outlandish by Star Trek refit standards), it suggests
Constitution-IIs were still a going concern up to the end of the 24th century.
I think they should be as common as Mirandas and Excelsiors in the TNG-era, and I don't like over-fitting because of production limitations and deciding Starfleet was made up entirely of eight classes of ships that had CG models available in 1997. If they put me on DS9HD, I'd absolutely tap the Eaglemoss archive and fill out the big fleet scenes with
Constitution-IIs,
Ambassadors,
Enterprise-B style
Excelsiors (heck, maybe even a couple
Excelsior-IIs, based on the registry numbers, they aren't that new),
New Orleanses (and the other BoBW ships, but that's my favorite), and anything else that could've been around then. I'd go mad with power, relabel the new Defiant with an -A, make new shots to replace the stock footage in the finale, and slip in cameos for the
Cerberus and
Enterprise-E. It's my daydream, I'll do what I like.
Rolling back to the main topic, the original
Enterprise had also been worked over in TWOK, which could've also contributed to the decision to decomission it as much as or more than the ship being anything but twenty years old, no matter how you count it.