That may be possible. Since there was a USS Dauntless NX-01-A/NX-01A, featured in the VOY episode “Hope and Fear.” And the VOY crew never questioned the registry belonging to another Earth ship. So maybe – for the duration of the Romulan War – the Dauntless became the flagship of Starfleet during that time, and took the registration on NX-01 for itself. Seems like something the MACOs would name their ship. And that would mean they outfitted that ship with MACO weapons instead of Starfleet weapons - which would help explain the various phaser streams seen during TOS, since they were merged with Starfleet after the war. Plus, we'd have to also consider that the Vulcans, Andorians and Tellarites had their flagship for the war too. As for the USS designation, the USS can be seen on a screen for both Enterprise and Columbia during the episode “Divergence.” Which means its considered a United States Ship at this point in time. As is the USS Spartan from the Star Trek: The Beginning script. In which case, those secret missions would have been done by the MACOs, or another branch of the United Earth military. And also means they handled atomic weaponry as well. Which would mean Archer and his crew would be doing other things with their time while the war was happening. Like vacationing on Risa.