1) A fairly obvious in-universe rationalization: the planet was surrounded by an asteroid belt that made travel by small craft faster than travel by large starships. I mean, we directly learn there's at least that one asteroid there.
But the presence of that one asteroid seems to come as a surprise, after some warp travel away from Epsilon Canaris and "towards Gamma Canaris", that is, an interstellar stretch. And when Scotty gets there, he sees seven thousand asteroids that definitely surprise him and Sulu, who says "what appears to be an asteroid belt". Postulating a belt also around Epsilon Canaris would have to be ad hoc.
2) An in-universe rationalization in line with TNG era evidence: Epsilon Canaris was one of those stars that interfere with warp drive so that a starship is rendered just as slow as a shuttlecraft, making it vastly preferable to have the latter rather than the former stuck in that tar. We see a lot of this in the TNG era shows: a shuttle is dropped off some distance from a destination and proceeds there, typically under impulse, while the mothership proceeds elsewhere. And we see that certain places such as Bajor and Earth are warp-adverse, at least on certain days (subspace weather!), with the characters commenting on how it's "risky" to warp in there, or then flat out refusing to warp even though it's a major emergency.
This is undermined by the shuttle here being at seemingly effortless warp, though (the Companion approaches at explicit warp, then is said to be "staying right with" the shuttle, which would not be a comment worth making if the shuttle were slower than the Companion).
3) The trivial explanation: two warp-capable vessels racing towards each other get the job done faster than one. If Kirk and pals had merely waited on the surface of Epsilon Canaris III, they would have met with the Enterprise later.
But that would require Kirk and his shuttle to be on the planet to begin with. Yet Hedford's medical emergency appears unexpected, and the impression we get is that she is not on the planet as part of Kirk's mission there (she comments derisively on "that ship of yours", as if never having been there before). Rather, Kirk was summoned there after the emergency developed.
So if Kirk is rushing to bring help to Hedford, why does he abandon his ship halfway through and switch to the slower shuttlecraft? Especially as the necessary and sufficient help explicitly specified is the sickbay of the ship, rather than some hospital far away?
4) The political explanation: Scotty in his warship has to stand off so as not to worsen the crisis.
But Scotty subsequently does move in, "backtracking" the shuttle trip. Does Epsilon Canaris III go down in flames as the consequence of this?
5) The more urgent mission explanation: Scotty had to be somewhere else. But this is weakened by the fact that Hedford only needs to get aboard the ship; once she's aboard, everything is fine again. So Scotty really should pick her up, then rush to his urgent assignment, and complete it while Hedford recuperates (he supposedly has completed it by the time of the rendezvous that the shuttle here fails to make, and everybody agrees this is soon enough to save the peace).
I guess our best bet is mixing a bit of 3, 4 and 5, with extra sauce and fries. Yes, Hedford has a medical crisis. But until McCoy gets there, nobody realizes it requires the resources of the starship's sickbay. And since the urgency was unknown, Kirk decided to go in using an inoffensive shuttle, sending Scotty to deal with what he considered a more important use of the starship. For greater effect, Kirk himself went along, to placate the Canaries and perhaps also Hedford.
With the alarming diagnosis completed, the shuttle and the ship now have to rush towards each other. It's a tight call, and the delay in fact kills Hedford. "Luckily" for her, a space monster possesses her body for use as Zephram Cochrane's sex doll, and war on Epsilon Canaris III can proceed.
Timo Saloniemi