But I don't equate Kirk's ship exactly to the modern Navy. IMO the Enterprise was more equivalent to Captain Cook's ship in the 1600 or 1700's Navy. Where the Captain was on his/her own making decisions, not like the modern Navy where the Captain can pick up a phone and dial up their superiors at any time.
If there were English colonies that needed supplying along the way then Captain Cook would do that maybe.
The point is, it's mission was pretty varied. If you look at the first season you see that
1 The Corbomite Maneuver- Mapping (exploration)
2 Mudd's Women- "Traffic control" (space cop)
3 Enemy Within- Specimen-gathering (exploration)
4 The Man Trap- Routine medical examination. (Federation business)
5 The Naked Time- Picking up a scientific party. (Federation business)
6 Balance of Terror- Patrolling the Neutral Zone. (Military)
7 What Are Little Girls Made Of- Looking for a lost scientist (Federation business)
8 Charlie X- Transporting Charlie to Colony 5 (taxi service)
9 Dagger Of The Mind-Cargo drop at a penal colony (Federation business)
10 Miri-Responding to a Earth style distress signal beyond explored space. (exploration)
11 The Conscience of the King- Unknown, but rerouted to provide transportation for the Karidian Company. ( taxi service and space cop)
12 The Galileo Seven- Delivery of emergency medical supplies/scientific research. (Federation business)
13 Court Martial- Layover for repairs/ inquiry about the death of Finney. (Starfleet business)
14 The Menagerie- Docked at Starbase 11 in response to a request to divert there. (Starfleet business)
16 Shore Leave- Exploring a new planet/shore leave. (exploration)
17 The Squire of Gothos- Delivering supplies to Beta Six.(Federation business)
18 Arena- Responding to a call from the Earth observation outpost on Cestus Three (Military)
19 The Alternative Factor- Scanning a planet and then off to Starbase 200 (exploration)
20 Tomorrow is Yesterday- Heading to Starbase 9 for resuppply. (Starfleet business)
21 The Return of The Archons- Looking for the Archon (Starfleet business)
22 A Taste of Armageddon- Opening diplomatic relations with Eminiar Seven (Federation business)
23 Space Seed- Salvage? (Starfleet business)
24 This Side of Paradise- Checking on Omicron Ceti Three (Federation business)
25 Answering a distress call on Janus Six (Federation business)
26 Errand of Mercy- Preventing the Klingons from establishing a base on Organia. (military)
27 The City on the Edge of Forever- Scientific investigation. (Exploration)
28 Operation: Annihilate!- Trying to save Deneva from mass insanity. (Federation business)