It's certainly how the Millennium Falcon works. Luke, get on those guns! R2 see if you can figure out why the hyperdrive isn't working!
Note that the doesn't tell R2 to shoot the guns, and Luke to repair the hyperdrive. Ever. Astromech droids are built to do repairs in space, help manage power distribution, and to help in hyperdrive navigation. That is literally the reason they are built. Their raison d'etre. He's merely telling people to do the things they have some competence at.
There are some fairly common roles in small group tactical combat in our games. There's no reason to think that ship-to-ship combat should not have roles based off of skillsets and abilities, too. So, while maybe the tactical officer can go and reverse the polarity of the neutron flow of the engines... they probably shouldn't. And maybe the doctor doesn't have a lot of training with the ship's weapons.
What you can do is separate the ground-combat roles from the space-combat roles. The guy who's good at shooting and stabbing on the ground.. maybe they're the Communications officer in flight, and so on.
Any suggestions?
Ashen Stars has a space combat system that is designed to allow everyone to act over the course of a combat. However, while it has tactical choices, they are abstract, not based off concrete ideas of physical positioning or the like. From what you've said, I'm not sure you'd find it satisfying.
A little while back, I played a Savage Worlds knockoff of Star Wars that had a reasonable set of starship combat mechanics that passed the baton around the team a lot. I don't know what exact ruleset the GM was using, however.