In Journey to Babel when Ambassador Sarek became critically ill, why didn't they send him home or send for a Vulcan medical expert? Since they were hosting many prominent political figures from many races, it seems strange that they did not have much medical support for a race so common as Vulcans (who are part of the Federation).
In the episode transcript, McCoy makes two things clear:
- The operation is an urgent necessity. Sarek's heart appears to be failing and it is a peculiarity of Vulcan physiology that this can't be ascertained (let alone corrected) with sensor readings alone.
MCCOY: As far as I can tell from instrument readings, our prime suspect has a malfunction in one of the heart valve. It's similar to a heart attack in a human. But with Vulcan physiology, it's impossible to tell without an operation. Mrs. Sarek, has he had any previous attacks?
- That he (McCoy) has reasonable hands-on experience of Vulcan general practice medicine as well as more extensive theoretical understanding of their organs and structures.
MCCOY: Mister Ambassador, I understand you had retired before this conference was called. Forgive my curiosity, but as a doctor, I'm interested in Vulcan physiology. Isn't it unusual for a Vulcan to retire at your age? After all, you're only a hundred and two.
MCCOY: Plus the fact I've never operated on a Vulcan before. Oh, I've studied the anatomical types. I know where all the organs are. But that's a lot different from actual surgical experience. So if I don't kill him with the operation, the drug probably will.
In the end, his surgical skill isn't the main barrier to success but rather the limited amount of compatible Vulcan blood. Not only does the operation in question require heroic quantities of blood but there's the added complication that Sarek has a relatively rare blood type:
SAREK: My blood type is T-negative. Somewhat rare, even for a Vulcan.
Canonically, Spock was the first Vulcan in Starfleet. Starfleet was, at this point in Trek history, very human-oriented. It is not surprising that Enterprise was ill-equipped to deal with a medical emergency involving a species not generally found on a human spaceship.
In addition to what Richard said and to address the 'why not go to Vulcan' idea, the plotline regarding Journey to Babel involved 1) one of the ambassadors being assassinated right before McCoy discovered that the urgent surgery was needed and 2) the appearance of an unidentified but hostile spacecraft.
Very shortly after Sarek's condition was realized (he was questioned as a possible suspect re: assassination and that was how his condition got incidentally discovered) and he was escorted to Sickbay, the situation regarding the unidentified spacecraft escalated.
Hostile craft sharing space in neighboring space coupled with an ambassador assassination and Kirk's own injury all kind of happening at once was a lot on its own.
If Sarek's condition had been discovered earlier and BEFORE everything escalated, then perhaps a trip to Vulcan would have been more feasible.