It probably feels less broad because he's playing Kirk imitating these guys
Precisely this.
Tribbles builds to the big pile on Kirk, but that's pretty much the only over the top gag. The rest are all in-character bits.
Very true. I think the difference for me is just how Shatner plays several of the bits. He never (almost never?) throws away a laugh line the way that he does in almost every other episode that has funny bits but isn't a comedy.
"I have never questioned the orders or the intelligence of any representative of the Federation."
Waaaaaaaait for it.
"Until now."
Never let Shatner know that he's expected to be funny. It reminds me of the difference between Leslie Nielsen and Lloyd Bridges in Airplane. Bridges kept trying to be "funny" and they had to tell him to stop. Nielsen figured out very quickly that the funniest thing he could do was play it totally straight.
Everyone else plays their characters no differently than at any other time. Nimoy knows that Spock being just as deadpan as always is funny. Kelley is as funny as he is anyplace else. Walter is the plucky comic relief anyway. He's practically Guy Fleegman. And Doohan? I mean, imagine Doohan trying to be MORE broad than usual?
So the variable is Shatner.
Not to cross, or at least bend the streams, but I think More Tribbles More Troubles is less of a comedy than TTWT is. The couple of overt comedic beats are just echoes of TTWT (and probably contribute to it's reputation as a total rehash) but otherwise it's an adventure story that happens to have Cyrano Jones in it.
Of all the episodes to get a Fotonovel in the 70's, man was I disappointed this was one over the more visually interesting Doomsday Machine or Corbomite Maneuver.
I can't argue with that.