Today is a day when I feel terribly tired. I'm not ill, but I could sleep all the day, though I had enough sleep last night.
This is not a problem if you have an office job where you can probably be not as productive than on other days. But what about pilots? They are expected to have a high level of concentration and vigilance during the whole flight?
If a pilot is not rested enough to safely operate the aircraft, they should not fly. [...]
Well, "should not" sounds more like advice than a rule.
You may say that there are always two pilots on (not so long) commercial flights, but then you can also start with one faulty engine, because the second engine still could bring you to your destination. For me, this breaks the safety concept.
So, can a pilot go to a doctor and tell, "I'm tired today so I can't work"?