Or in other words, what's the difference between a sick day and "turn round and go home and back to bed, on the basis that you're likely to do more harm than good if you continue trying to do work" from the point of view of "getting back into a mental state capable of work after three weeks of holiday"?

A fair point! I think in my head there's some difference between coming in to work, trying to do something, and giving up and going home again, and on the other hand just staying at home in the first place. Probably the former has at least some value in getting me back into the rhythm of getting up early and going to work. But it's pretty borderline, so you have a point regardless...

Perhaps I should also note that I've taken sick half-days in the past when I came into work but discovered part-way through the day that I wasn't up to the task - further blurring the distinction.

Essentially, it then comes down to whether you don't feel up to working because of general listlessness or because of, say, a fever.