That's a thought I'd had many times watching TOS: what's happening on the other ships? Who's making first contact with a cool new species? Who's discovering secrets of some lost civilization? Who's sticking it to the Klingons and/or the Romulans?
If you watch some of the TOS episodes, something else seems to be there too: Ever notice that it doesn't sound like the Enterprise is the best ship, or that Kirk is the best Captain? No one's ever like "Oh wow, Kirk, captain of the best ship in the fleet is here!", but more like "Oh finally! they sent someone! fix this!"; it's almost like they are just some space rent-a-cops that HAPPENED to discover a threat to the galaxy, and got lucky stopping it. In that regard, he's less this:
And more this:
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"Yes, it's the Apocalypse alright. I always thought I'd have a hand in it"
Professor Farnsworth