So, the chair is finished and getting a lot of use in the loggia. It isn't perfect, but a fairly successful conversion anyway.
The tops of the legs were rounded with a chisel and block plane. Eveything got a light coat of raw linseed oil and beeswax, hopefully the birch will darken to match with age.
The upholstery job was easy, with foam from an IKEA cushion (IKEA gets a lot of hate, but 19 kr for a chair's worth of foam cushioning plus cotton for two shop rags is a great deal.) and some nice patterned blue fabric. The fabric was far and away the most expensive part of this project, but there's enough left for a throw pillow or two. The foam was contact cemented to the plywood, and the fabric drawn and stapled in place.