I used to play Starcraft Brood War at low-levels of competitiveness (maybe ~120 APM or so, D-rank Team Liquid. Very low end of competitive). I also enjoy Factorio and Pikmin, so I'm no stranger to multitasking.
Ideally, you want to have a game where so much is going on, that you *NEED* to split up the tasks to multiple teammates. I don't know, maybe have each of the players doing an "Overcook'd" style minigame, trying to do whatever their sections of the ship need to get done.
So the idea is that each player needs to actually focus their mind on their minigame, but _ALSO_ needs to juggle the discussion with the captain / bridge. I can't imagine how a game like that would work, but... sitting around calmly in a "Star Trek" style bridge isn't very fun IMO.
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I think we have an issue of expectations vs gameplay. Anything that requires a team of multiple humans would necessarily be incredibly hectic. (See DOTA, Counterstrike, Overwatch, and other team games).
But on the other hand, there are people who want to role-play Star Trek scenarios, which isn't... very realistic from a team perspective. Just as combat in Hollywood is unrealistic (1 vs 5 fight is really a sequence of 1 vs 1 fights 5 times in a row: every enemy waiting their appropriate turn with the hero...) Star Trek scenes progress logically and simply, for the audience to follow. Its almost the opposite of the chaos that requires a command structure of any kind.
"Archon" / Team Melee from Starcraft 2 / Starcraft Brood War was also a good bit of fun, since two humans playing 1-commander requires communication (make sure both players are going for the same build), and you naturally figure out how to split up work with your teammate.
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I guess a more appropriate "Bridge" game is "keep talking or the bomb explodes". At least for people who want "my" style of hectic team-based game with difficult communication going on. If someone can amp up the "difficulty" of bridge games up to this level, it'd be more interesting to me.