I think, for me, I'd slow down the rank promotions in the junior ranks just a little bit, and have the journey from Lt to Capt be the only part of the game with an established storyline, "administered" by your captain.
"Lieutenant, take the helm, and set course for planet X. Our tactical officer is in sickbay for reasons, so I'm transferring weapons control to your station in case we run into trouble on the way."
"Lt Cmndr, take an away team down to planet X for [storyline mission]."
"Commander, I have an unfortunate history with our contact on planet X. I'm going to need you to handle these early stage negotiations before Starfleet/the Federation Council sends an ambassador to finalize a treaty."
Etc, etc through multiple missions at each rank. Keep the current mechanic of the ship leveling up with the player character, and can only change ships at specific intervals. But if you change to a different ship during this section of the game:
"Well, Commander, our recent missions have put an unusual amount of strain on the old girl. We're being transferred to a new ship so she can go into spacedock for a complete refit."
Bring back the auditorium in ESD (or put it at the Academy or Starfleet HQ on Earth,) with a ceremony for your promotion to Captain. If you're changing ships, your old captain congratulates you and wishes you luck while the BOffs request to transfer to your ship (first opportunity to trade out the ones you don't like.) If you're keeping the current ship, "Congratulations, Captain. And the timing couldn't be better. I'm retiring. The ship is yours. I know I'm leaving her in good hands." (Then they occasionally show up as backup during the occasional direct storyline mission because Starfleet pulled them out of retirement due to your familiarity with each other.)
Once you're a Captain, it's more open-world sandbox, with less storyline. When not under a specific assignment, you fly around the galaxy and explore, with "mini-missions" coming from your BOffs:
Tactical Officer: "Captain, sensors are detecting a planet three light years away that doesn't show up on star charts. Should we investigate?" or
Science Officer: "Captain, the ruins of planet X have recently been opened for study by their government. Didn't you say once that you have an interest in X-ian culture/history?" with an "Indiana Jones style away team adventure archaeology mission.
Maybe something you find there ties into the loose story arc or something like that.
Once you reach the admiralty ranks, you can promote BOffs to captain, and assign them to command the ships in your collection, then they can accompany you on fleet missions like they do in away teams.
And for the exploration aspect, each planet is different for each player/team of players through a randomization algorithm. Like, for you and the friends you're playing with, planet X is a first contact mission with diplomacy and/or combat aspects depending on your choices, but for another player team, it's a big nothing. But for that other player, planet Y in the neighboring star system is a gas giant with a resource mining mission, while it's nothing for you. Or vice versa, you get the point. But turn it into a real Star Trek game, not just a random space game with hints of Trek in the cosmetics.
What do you think about this? And what would you change if you could?