Well now I'm not sure we're even talking about the same show anymore.
He's not wrong. The opening monologue is more poetry than fact. The TOS Enterprise did more than explore. It transported diplomats, checked up on colonists and scientists, captured criminals and defended the Federation.
Lets look at season one
01 The Man Trap: Physicals for scientific party.
02 Charlie X: Passenger transport
03 Where No Man Has Gone Before: Exploration
04 The Naked Time: Picking up a science team.
05 The Enemy Within: Exploration
06 Mudd's Women: Policing the spacelanes/Passeger transport
07 What Are Little Girls Made Of?:Checking up on a science team
08 Miri: Answering a distress call. Exploration
09 Dagger of the Mind: Supply delivery
10 The Corbomite Maneuver: Star mapping. followed by a military confrontation
11/12 The Menagerie: No actual mission/Court martial
13 The Conscience of the King: Passenger transport/Criminal investigation
14 Balance of Terror: Military
15 Shore Leave: Exploration/RnR
16 The Galileo Seven: Supply delivery/exploration/rescue
17 The Squire of Gothos: Supply delivery/exploration
18 Arena: Military
19 Tomorrow is Yesterday: Resupply/getting back "home"
20 Court-Martial: Military legal proceeding
21 The Return of the Archons: Exploration
22 Space Seed: Rescue/Recapturing the ship.
23 A Taste of Armageddon: Diplomatic transport/Military
24 This Side of Paradise:Checking up on a colony
25 The Devil in the Dark: Colonial defense/Monster hunting
26 Errand of Mercy: Military
27 The Alternative Factor: Exploration/Rescue/Criminal pursuit
28 The City on the Edge of Forever:Exploration/Rescue
29 Operation-Annihilate:Military