This sounds similar to STO's Duty Officer missions.
While you normally play as the "Captain" of the ship with one to four pet bridge officers, you can queue up these duty officer assignments carried out off-camera by your (sometimes) hundreds of available crewmen. You can collect duty officers with different stats to improve your chances of a successful mission and gain all sorts of stuff from in-game currency to consumable items. STO's system also lets you gain some XP. If you keep DOff assignments always running you can sometimes gain a level while offline.

This sounds a heck of a lot like the [url=https://sto.gamepedia.com/Duty_officer]Duty Officer[/url] system used in Star Trek Online.

Duty Officier missions sound like the crew mission in SWTOR. Its a series of tasks that usually revolve around collecting rare materials for crafting. But the player doesn't have to actually do anything other then send a crew member.

Sounds like work for to me. Others may like it. The crew missions in swtor were for crafting or getting materials for crafting (mostly). I would spend time managing my crew missions, swapping from toon to toon to gather/craft. In the end it was good for my in game pocketbook and I was supporting my guild with such stuff. But it was also a lot of time where I wasn't chopping up bad/good guys with a light saber. So it kinda became like work. So if I did it, I wouldn't do it too much and if it was "required" (not that the OP was supporting that but this happens in MMOs), I would prefer that it wasn't.
I appreciate the possibility of incorporating story elements about sidekicks/henchpersons and if this unfolded as I went on missions myself that might be more interesting as it would add to my direct (not Meta) gaming experience.
The other issue would be balance in the game from any associated perks. Perhaps perks should be minimal in effect or have long reset timers. If not then perhaps consider it a tertiary power of some kind, where you have to make a decision about where your power's are distributed. Just letting my thoughts wander there.

The biggest strike against this sort of "crew mission" style of things is just the sheer multiplication of time that they "demand" as the number of alts you're playing goes up.

When I was playing Star Trek Online, I had 4 captains (2 Starfleet, 1 KDF, 1 KDF aligned Romulan). Simply doing all of the admiralty and duty officer stuff for ONE captain could easily consume a half hour of my play time. Multiply that by 4 and there goes an entire 2 hours of my life PER DAY. That basically wound up being hours of time in which the game was "playing ME" rather than me playing the game. Then if I wanted to do things like mine for Dilithium or visit all of the Fleet Holdings, tack on another half hour (or more) per captain to do those on a daily basis. POOF! There now goes an easy 4-6 hours of my life PER DAY simply doing "dailies" that have almost nothing to do with playing the game or advancing my characters through the game. I'm basically spending HOURS each day just running logistics on each of my captains, and for what? So that I'd be "rich enough" to finally at long last Play The Game eventually?

It got to the point where if I wanted to actually PLAY the game, I needed to devote an entire day off to doing so, something on the order of 12+ waking hours, just because there were so many ways to fritter away my time on a daily basis "doing chores" on each of my captains, rather than spending time having fun PLAYING THE GAME. Rather than going out and doing ground battles or space battles, I'd just log in to stand around and battle my way through menus and click on things in windows ... for HOURS ... every single day.

As has been pointed out repeatedly, at that point, you aren't playing a game anymore, the game is playing you. You aren't logging in to have fun, you're logging in to FARM. It stops being a diverting pastime and instead becomes another JOB.

And that was with just [][i]4[/i][/b] captains.
Now try to imagine the Treadmill Hell for anyone playing 10+ alts.

Yeah, no thanks.

To be honest, if I want to play a management game, I'll grab something like the Civilization series or a Master of Orion styled game to get that fix.

Duty Officers were [i]cool[/i] in Star Trek Online, when they were first introduced ... but then the system got extended and replicated (multiple times) to the point where it basically took over being able to play the game itself. Being able to load up 20 missions for your Duty Officers to go do, and then load up another 20 missions for all the ships in your "fleet" that you've ever owned (or bought from the game store) ... on every single character you're playing ... that turns into its own management game, all on its own, and its one that in effect cannibalizes play time from the main game so as to play the (no longer) "mini"-game on all of your characters.

So I'm going to lay down a marker for this sort of thing.

[b][i]IF[/i][/b] City of Titans winds up doing something like this, it would be a Smart Move™ to necessarily limit the number of slots for objectives to be running simultaneously. Rather than having 20+ slots per character (creating a sense of "obligation" to fill them all), only have like ... 4. By limiting the number of "mini-game missions" that you can have running in the background like that, you necessarily limit the amount of DISTRACTION from playing the actual MMORPG itself, while at the same time creating a prioritization system in terms of what you want to get "done" through the mini-game at any one time. Furthermore, you'd be able to set up the rewards of the mini-game as being something that can help prevent PCs from going dead broke on IGC or other resources, but by the same token it doesn't wind up being a cash cow farming resource either.

By limiting the scope (or reach, or demand, if you prefer) of the mini-game, you prevent it from taking over the Player's (precious!) time when the Player has a huge number of alts to tend to.