@williams_482@startrek.website invited contributors from the old Daystrom to repost some favorites, so here is one of mine.
Throughout Star Trek, but especially in TOS and TNG, we are commonly asked to be very stressed out about our captain being overruled or displaced. Regardless of whether the replacement does a good job, it seems clear that we are supposed to resent him simply because he is not the usual captain we have come to know and love.
A particularly striking example of this is TOS "The Deadly Years," where Kirk is aging rapidly and apparently going senile. This seems like a clear case where Spock should step in -- but a good chunk of the episode is taken up with the procedings to relieve Kirk of command. In the end, the inexperienced starbase commander who replaces him turns out to be a disaster, and the ship is only saved when a cured Kirk is able to come in and be his usual decisive self.
The most gut-wrenching example, of course, is Captain Jellico, who arbitrarily changes everything, criticizes the way Troi dresses, won't let Riker do his job -- and regards it as a foregone conclusion that Picard is dead.
I have seen several comments to the effect that the crew's response to Jellico is a little childish, and I think that's a clue to what's going on with this common plot. Namely, I believe that the captain is put forward as a father figure and that the displacement plots are speaking to a cultural anxiety about divorce. The replacement captain is the step-dad who always appears to be an illegitimate usurper -- and in the end, we get the fantasy outcome that mom and dad get back together again.
This may seem far-fetched, but the earliest TOS episodes do a lot of work to establish Kirk as a father figure (most explicitly in "Charlie X") and the ship as his wife ("The Naked Time"). This is more subdued in TNG, where Picard is awkward with kids -- but Picard's emotional distance completely fits with the "traditional" image of the father. Surely "Captain Picard Day" is something like Father's Day for the Enterprise children! And more broadly, the backstory of many Enterprise crew members includes broken families, alienation from parents, dead parents or spouses -- all factors that lead them to identify the ship as their true family (and invite the misfits in the audience to do the same).
Over the years, of course, our culture became less and less stressed out about divorce as it became more routine -- and so those plots suggested themselves less and less. In DS9, it is far from a dominant theme. I haven't rewatched in a while, but I don't remember even a single plot that hinges on someone taking over for Sisko -- when the Dominion takes over the station, the emotional focus isn't Sisko's lost command, but the loss of the station itself. [ADDED: I wonder if the fact that Sisko is the only captain who is presented as a literal father somewhat undercuts his role as father-figure thematically.]
And Janeway's command is never seriously disputed. Of course, in-universe you can say it's because she's so far away from the admirals, but symbolically, she's the mom -- and in a typical divorce narrative, it's never a question of whether mom will remain in place. The one clear example I can think of where the crew rebels against her authority is "Prime Factors" -- and their main rationale is that they believe Janeway's judgment is clouded by her obvious attraction to the leader of the vacation planet. In other words, the kids get restless when it looks like mom might have a boyfriend.
The theme of the displaced captain comes back somewhat in Enterprise, but to me it feels different. The issue isn't Archer being replaced by a step-dad -- instead, the problem always centers on Archer's masculinity. In "Hatchery," he becomes overly maternal toward the Xindi Insectoid babies, which leads to a mutiny. Similarly, in "Bound," the Orion Slave Girls compromise Archer's judgment with their aggressive sexiness. Archer's either becoming a woman or being dominated by one -- which calls back to the early episodes, when it could sometimes be unclear whether he or T'Pol was really in charge. Archer represents not a father, so much as an emasculated human race ready to prove itself -- a more reactionary theme for a more reactionary time (the early 2000s).