Some quick comments:
-stepping through the scene where Stone is viewing the extract frame-by-frame yesterday, the 8 in 1831 always looks like an 8 except once, so, at least on a 55" LED TV, it is 1831. The second digits below it that are 6s always looked like 6s.
-even if you say that only 17xx is "like her in the fleet", i.e, Constitution-class, there are still five
of the twelve* listed here, apparently laid up for repairs/maintenance.
-1697? Under the Jefferies' paradigm, the 97th "bird?" Of course, the one hundred numbers could have been assigned to the various shipyard in blocks, ie, SF gets 1600-1615, Utopia gets 1616-1630, etc. So a build order might be 1600, 1631, 1616, each yard finishing the first of its number block. That would allow for a smallish fleet (that TOS often suggested) but such large serial numbers.
-the Jefferies system does not have to continue into the 24th century. A switch to straight serial numbering could have occurred as the Jefferries system became too cumbersome to use. Such changes happen all the time IRL.