Perhaps because Sickbay is in essence the equivalent of a Hospital, last I checked Hospitals are manned 24/7.
To operate a 24/7 shift pattern, including covering mealtimes, you need about 6 shifts.
In the real world, Pitcairn has 1 doctor looking after the island's population, with outside help being days away. Same applies in Antarctica. If there's an emergency, the doctor will be woken up. If the doctor is incapacitated then you have to make do.