And was that 12 a static number? Personally, I like the large number of Connies in FJTM & SOTSF, and figure that a small number of them are outfitted for independent duty, as the Enterprise is during TOS. It wasn't, IMO, during the 2 pilots, but was refit to independent duty specs after WNMHGB.
So when Kirk brings it up, his ship is one of 12 or 13 with those specs and that special status. Before TWOK, it isn't anymore. Those honors have passed to other ships. As more ships get built, more of them can be independent duty.
Regarding the age of the UFP, I like the idea of the organization going back around a century, so my thinking is: In the 2160's the big 5 (Andor, Alpha Centauri, Vulcan, Tellar, and Terra) form an alliance: UN/NATO writ large. UESPA was the Terran branch of the combined military. Back when it was formed, it was an allied military, but over the last century, it's been molding each world's militaries into a single standardized model, and the Babel conferences have been slowly consolidating each world's sovereignty into the whole. Sometime in the 2260's, the work was complete, and the politics finally came together to turn the UFP from a non-governmental organization akin to the UN into a binding government ala the US federal government. The TMP uniforms are the first new issue of the (finally) consolidated Starfleet. Prior to that, Starfleet was a theoretical term describing the combined military services of all members, but in practical use, UESPA told Terran ships what to do, Vulcan High Command told Vulcan ships what to do, etc.
By the time Kirk brings his ship back after the 5 year mission, all those agencies have finally been folded into one, and just as he gets promoted, the uniforms change over - the new organization wants it's own uniforms, not one inherited from it's predecessors. Technical architecture has been pushed to develop the whole new aesthetic for the new service, as seen in the revamped appearances of everything in TMP, and perhaps that, as much as any technical capabilities, is why the switch from reactors in nacelles to reactor in the hull.