In at least one draft, Nogura actually has the line, "It's a captain's command," which made it pretty clear that he's the one making that determination, or at least the one enforcing it.
Interesting. I suppose it makes sense given that Enterprise
was a single ship and not part of a fleet of ships intended to stop the intruder. Had the vessel been part of an armada, it may have made sense for an admiral to command the fleet from a designated flagship as seen during Deep Space 9
But how often did we actually see admirals doing that in the TOS era? I recall Commodores having expanded responsibilities, but admirals were basically stuck behind a desk somewhere, and never on a ship. DS9 is over 80 years in the future from TMP, and in the intervening years, the Commodore rank disappeared and many other changes were made.