A shortage of riboflavin (vitamin B2) seems to be the most likely reason. Riboflavin is produced in the bird's gut, where it gets passed out in the droppings. When there's a dietary shortfall, chickens can sometimes resort to eating their faeces.
As it happens, B2 is a fairly unstable vitamin in the presence of light. Perhaps my current batch of chick starter was exposed to light while in storage or transport; or it may be that the parent hens who produced these chicks had a deficiency themselves. Layer diets aren't formulated to necessarily produce healthy chicks.
Luckily I have a couple of options for fortifying the chicks' feed. To start with I've started mixing some brewer's yeast with their usual ration (brewer's yeast is very high in B vitamins). Secondly I've begun adding small amounts of dried skim milk. Milk isn't recommended for birds, as they have no capacity to absorb lactose, but in this case I think the benefits outweigh any drawbacks. If I see evidence of diarrhoea I'll take it out.
Another useful tonic for chicks is a bit of yoghurt with grated carrot or apple mixed through their chick starter as a mash. This suggestion comes from chook forums, and it seems to work for me. The yoghurt not only helps with B2 but also provides probiotics. And you'll rarely see chicks turn their noses up at such a mash.
My chicks have just gorged themselves and are lying about happily now. Hopefully that will be the last of the serious poop eating!