Not necessarily...
Remember the command crew is also the people most appropriate for representing the ship (and the government/organization they represent) in all official matters. So it's actually just as, if not more, likely that the command crew of a large warship will have to go in person to deal with official matters. How would you feel as the Grand Poobah of some colony or undiscovered world to have a kilometer long warship show up and the representative they send to deal with you be some junior officer or enlistedman not authorized to make big command level decisions?
I think it boils down to mission. I think the problem is that not every mission you'd run in such a game would be important enough to send the captain down. Thus keeping the command crew involved in all the missions can become difficult. For instance, sure the captain may go speak with representatives from another planet that the Alliance is trying to woo to their cause, but he's not likely to go on a mission to infiltrate a Imperial base. Running the command crew will obviously change up the styles of missions you can run and I think in some ways that can be limiting.
That's on the Campaign Design side of things.
If you're going for a Trek vibe, then of course the captain can go on all kinds of silly missions. After all, that's how "There are four lights!" came about.
If you want to go more realistic, then the Captain and command crew can do a lot... but the covert raid on a bioweapons factory is really a job for the special ops team you carry aboard... and it's time to work out the multiple character option...