Maintenance is staggered because there is only so much room for people to work in inside the engine room.
For most ships, it would be cheaper for the company to pay for the crew than lose out on shipping contracts due to being in port for maintenance.
Engineers can detect a lot of problems that machines can't, be it unusual smells, sounds, vibrations, temperatures, etc. Sure there are probes that can detect vibrations and temperatures, but they would have to be placed everywhere (not just the hundred or so usual spots), and then they have to be programmed. I use a handheld device to measure vibrations on running machinery. It's only as good as it's programming, and it's not that good.
Additionally, hoses burst, o-rings leak, filters need changing, etc. We don't have machines today that can fix those problems on the fly.
My last ship I was on was Diesel Electric. It was incredibly automated, yet still we had MAJOR problems because some circuit boards died.