I just want to make a general post about two things I think are getting conflated:

#1: Actual military protocol regarding what is and what isn't the captain's responsibility

#2: What a non-military captain's responsibility would be in this specific situation

For #1, yes, in the military basically everything is the captain's responsibility, and even if it's not official, it doesn't matter, everyone considers it the captain's responsibility

For #2, the military is a good example, but in this situation I have to agree with grey_wolf in that yes, the ship and the crew are obviously the captain's responsibility, and the captain is also responsible for the crew's morale. BUT, the moment a crew member stops considering the captain as The Captain, that crew member can't be under the captain's responsibility because that crew member doesn't acknowledge the captain's authority.