It's more plausible that a science officer would even know what to do to fix the warp-drive rather than Kirk. Cinematically, having him just abandon the bridge at that moment for the turbolift would have looked weird. Instead you have the iconic moment of them turning to see Spock's empty chair (Spock having left when nobody was watching) and Spock rushing down to engineering. Kirk wasn't just going to leave without anyone noticing. Also, unless Trek II had immediately used the Genesis device to resurrect Kirk, by having him make the sacrifice would have pretty much ended the Trek franchise right then and there. While it's true that the cast and crew always produced each film as if it could be the last, there's no way they would have ended it with their hero dead. Losing Spock was as bold as they could be in the writing.
When you add it all up, all the plot elements in Wrath of Khan click together very nicely.
The only person who should know as much about how his starship runs than the captain is his chief engineer. I'd be surprised (and disappointed) if Spock knew more about how the Enterprise worked than Kirk did. He may know as much (which not all science officers would have to know), but not more. Kirk should've known what needed to be done, too. But he seemed paralyzed. He couldn't "corbomite" himself out of this one. Spock knew what he had to do before Kirk did, and he acted.
And of course they could've killed off Kirk. The next movie would've been about his resurrection, not Spock's.
By the way, they "lost Spock" not because it was the boldest thing they could write, but because Nimoy agreed to be in the movie only after Jack Sowards promised him a death scene. Spock wasn't even in Bennett's first treatment of the story because Bennett was sure Nimoy wouldn't do it. The original death scene was supposed to occur early in the story, but it kept getting pushed farther and farther into the story until Spock was basically in the entire movie.
Spock was Captain of the Enterprise in TWOK.
IMO it wasn't a real ship with 'real' missions, just a training ship all under Spock's command.
I know I know he gave command to Kirk. IMO Kirk was in command of the mission. The ship, the crew's compliment were Spock's responsibility. The tactics, the negotiations were Kirks.
Spock knew the trainees (to some extent) manning this Enterprise. Kirk didn't.
Spock wasn't science officer in TWOK and if Kirk as captain should know the ship better than anyone (except the chief engineer) so should Spock.
And since Spock worked on the ship and Kirk no longer did perhaps Spock would know the ship design better.
And perhaps it needed Vulcan endurance to fix the ship in TWOK. Scott looked like he had tried and failed.
When Spock turned command over to Kirk, the responsibility for everyone and everything on the ship and the ship itself transferred to Kirk, too. It's the nature of command. You can't split responsibility. It rests alone with the ship's captain. Spock went back to the science station on the bridge and filled his old roles as science officer and first officer.
I did say Spock certainly knew the ship as well as Kirk. Since it apparently became a training ship, it's unlikely it would've had any new equipment on it Kirk wouldn't be familiar with (unlike in TMP, where the refit with new technology meant Kirk had to catch up).
I'll grant you that Vulcan endurance may have given Spock an advantage in the situation, but that doesn't mean the thought of getting the mains back on line the way Spock did it couldn't have crossed Kirk's mind to at least try. Instead he stood there on the bridge and looked helpless.