I'm genuinely curious how there's so much work sailors have to abide by a schedule like this. Is this only in active war zones or is it always like that?
There is an enormous amount of work to do on warships (in my experience); the following doesn't apply everywhere always, but all of it applies sometimes and some of it applies all the time.
If you're in engineering, kit is constantly breaking and it is never all working; you'll never catch up, so it's just a question of what you decide to leave broken right now and what you need to fix. The things that are working need constant attention to make sure they stay working; you'll never get them all checked, so there's always something that definitely needs checking really quite soon.
In warfare branch(es), as in all branches, the manning situation is never right; there are always gapped billets, requiring other people who don't quite have the right skills and experience having to share that extra workload amongst themselves and to cover extra watches. Going to defence watches or the like (six on, six off, or even four on, four off) is a permanent mindfuck that leaves everyone as a kind of zombie, skating through on routine.
The kit is often old, and it's all pretty custom, and so much of it is painfully delicate. This ship doesn't like warm water, that ship has a habit of not being able to power the radar all the time, damn comms constantly up and down. With relatively new classes of ship, every one of the first dozen is basically an experiment, so the new kit refuses to work properly with each other, or maybe just plain at all, and the original purpose the ship was designed for is long gone and what was a cold water submarine hunter meant to last a week on its own before being back in port is now a warm water patrol vessel out for months at a time, constantly stressing systems that were meant to have a steady supply of cold water cooling and were never meant to be active for weeks at a time.
Every ship and crew will habitually be asked to do more, with less. Harmony time vanishes, billets get gapped, refits get put off, maintenance cycles become less frequent. It stops when ships basically cease to function.
Sometimes it's a wonder they don't sink upon leaving harbour :)
Yes most surface warships have at least four lookouts on every watch to visually look for vessels and other hazards. That system works well as a safety backup for radar as long as they pay attention and communicate effectively.
It's always like this, or worse. Active war-zones are a bit different if we are sailing through "hazardous" waters. Usually on top of work and normal watches, you'll man up some different watches since the crew will have to man extra guns.
Some divisions are different, but a majority of sailors have a pretty solid stream of work. Maintenance of a ship that is close to 20-30 years old is a constant battle. Hell, even a ship that is 5 years old requires meticulous care.
Deck division is responsible for maintaining a majority of the "skin" of the ship. So, a lot of painting and preservation. We are also responsible for all life-saving equipment on the ship: Life preservers, life-boats, any number of RHIB (Rigid Hull Inflatable Boats), life-lines, etc. A lot of this time is spent supervising extremely under-qualified and unmotivated individuals.
It's almost akin to teaching an unwilling 5 year old to read, and then every single day constantly monitoring his progress. As soon as your comfortable that he or she can read without supervision, they are transferred to another division, or are taken off the ship.