PULSAR: Lost Colony (its in ozytons list)
I'd say its the closest to a star-treky style game, with some random-ish quests that are either exploration based, or one of the factions based[WD Corp-Evil Corp Empire Faction, Colonial Union-Startreky faction, Gentleman Alliance- Pirate Faction, Fluffy Biscuits Company-Biscuit based faction], or bounty hunting... with both ship on ship combat, boarding combat and away missions (you can capture ships via boarding, and even switch to them, but you cant say, pilot 2 ships at once).
It's a 1-5 player game (i suggest atleast 3 real people, the ais are ok but not amazing) and get to play Captain, Science officer, Chief Engineer, Pilot, Chief Weapons officer. The contents a bit lacking (i doubt you'll get more then 20-25 hours out of it), but it does have a bit of progression, both in ship upgrades (which can be wrenched from a captured enemy ship if you have a trackerbeam thing or traded for at a station), and the `chaos factor` for the galaxy gets higher making some extreme events happen.. you ship WILL be destroyed eventually.
The jobs on the ship do have different stuff they can do, different consoles n stuff, which is nice. I personally find pilot the most boring... the science officer can launch and defend against hacking, handles scanning/launching probes and can heal people, the engineer handles the power output/heat of the engine and balances the load on the systems, even turning subsystems on/off for a bit more power (Who needs lights or oxygen on a ship when you could just have a bit more power to the shields!). The weapons guy pretty much controls the missile launchers/laser turrets on the ship, and has skills also for being stronger at in-person guns too.. the captain is unspecialized, and have lots of bonuses for the entire crew... The pilot gets bonuses to.. piloting..
It's got some little touches here and there that are cute, you can also play a random dice game in the ship lounge while your captain yells at the pilot on the bridge... Some ships have hidden places to hide smuggled goods so the scanning drones wont notice them.. theres a decent amount of ships, all with their own layouts, some have special stuff, and start with different focuses of equipment (or outright special equipment like the cloaking device) so the ships play a bit differently, some can literally hide from scans, some are more like big destroyers, ones literally a biscuit delivery service.. You can on purpose or accident blow up your ship by redlining your ship-engine-core, including launching it out of your ship last-second ala startrek to basically act as a nuke. ..Theres also nukes, they are rarer though. They recently added two new `species` you can play/fight/etc with, on top of normal humans, one of which is literally robots, the other a alien, all three species have a significant perk/downside.
Games still in EA, and its a small team, they do put out new patches every couple months, with a few bigger content patches a year. Of all the games here, it's perhaps the best ship crew experience i had, but i can also understand if people don't want to buy a EA game, and rather wait a year or two until its `done`.