Would that have been feasible (in terms of the captain being a human being that needs rest) on a ship like the Fitzgerald that operates pretty much 24/7/365 in busy waters?
I never served aboard a "small boy" (a.k.a. "tin can" a.k.a. destroyer), so I don't know what their operational tempo is like. On the carrier where I did serve, if we had extended up-tempo ops, there'd be times when the most-senior officers other than the CO and XO would take turns as "command duty officer" on the bridge as a backstop for the more-junior officer-of-the-deck (OOD), so that the skipper could get some rest. Of course, a carrier has lots of pretty-senior officers — small boys, not so much.
And to be sure, it's not quite apples to apples. An aircraft carrier normally spends most of its time in the open sea: To launch and recover aircraft (other than helos and vertical take-off and -landing aircraft), the carrier needs miles and miles of room to steam at high speed into the wind without having to change course for traffic. (If you're steaming into the wind at, say, 30 knots, the aircraft being launched or recovered get a head start in staying above stall speed.) Depending on the circumstances, even in peacetime a carrier might have one or more weapons-loaded aircraft in alert status, ready for launch on comparatively-short notice as a combat air patrol (CAP). When that's the case, the ship would need to be away from traffic, in case it had to quickly turn into the wind to put up a CAP. That's why, except when entering or leaving port, or transiting, carriers usually stay away from highly-trafficked areas.
And to be sure, it's not quite apples to apples. An aircraft carrier normally spends most of its time in the open sea: To launch and recover aircraft (other than helos and vertical take-off and -landing aircraft), the carrier needs miles and miles of room to steam at high speed into the wind without having to change course for traffic. (If you're steaming into the wind at, say, 30 knots, the aircraft being launched or recovered get a head start in staying above stall speed.) Depending on the circumstances, even in peacetime a carrier might have one or more weapons-loaded aircraft in alert status, ready for launch on comparatively-short notice as a combat air patrol (CAP). When that's the case, the ship would need to be away from traffic, in case it had to quickly turn into the wind to put up a CAP. That's why, except when entering or leaving port, or transiting, carriers usually stay away from highly-trafficked areas.