I've done some more research. It appears (according to IATA Travel Centre, which I think is the same content as Timatic) that Chile is unique on insisting on this requirement. Peru, Ecuador and Argentina don't seem to have this, and Bolivia has an exemption for foreign minors who have been in the country less than 90 days - which seems a very sensible clause, so they don't scare off genuine tourists.
I have also sent an email to the Chilean consulate, but yet to receive a reply.

Looks like it may be Galapagos rather than Easter Island this summer.