These movies have definitely lowered the bar unnecessarily low.
set that bar in 1966 during "The Naked Time" when during a critical orbit both the Chief Engineer and his Assistant (the only two on duty mind you) left because they were both called to the bridge.
That and Engineering only has one door...
Lol - Naked Time might squeeze by on the notion that even if not obvious, those crewmen might have been affected by the virus. I do prefer to view TOS & TNG THROUGH THE PRISM OF THE TIMES THEY WERE MADE. That does make some elements look silly. I'm sure if they could have afforded to film in a brewery every week, they would have.
I agree with the comments about the difference between Kirk and Spock's solutions to the damaged engine. Spock, while not an engineer, has sufficient technical skill to effect repairs and greater resistance to radiation than the other engineers on hand. He is the logical choice for the task despite there being other engineers nearby.
Kirk's problem is that he promoted an ensign to chief engineer so they're all off doing something else... evacuating the ship maybe? And it really isn't clear why they have magical glass and inspired transporters but no radiation suits or robotic drones - technologies that we have in the 21st century already, along with greater equality of the sexes.
I can see why they needed Harrison. "Our people keep dying when fixing our engines. Please tell us what to do!" "Errr... make me bombs - and be quick about it."