The purpose of the lining is to increase gluing surface area at the edges, and while on violin family instruments this is largely because the sides (ribs) are so thin, in guitars it is mainly because of the intrusion into the edges of the bindings that cut a swath through both the sides and the plates.
The pic above shows my linings in the process of being shaped, first with a plane and then rounded with a scraper. These are from an unbelievably beautiful stock of very old (and old-growth) New Zealand Kauri that I have had for years. It was part of a fit-out in the 1940's- mainly cupboard shelving that was being discarded in a building up-grade. It is resonant, absolutely even in density and silky smooth yellow to purple with sparkly accents when cut on the quarter. I love this stuff to bits and really enjoy lifting it to the status of precious jewels when using it. (Kerfing it is a bit annoying really...I'm caught between honouring traditional technique and honouring magnificent materials)
You might notice the grain orientation in the braces. The grain is vertical, and while there has been argument over whether vertical or horizontal grain is strongest, there is general agreement, I think, that in terms of stability, verticality is king. In any case this is the way it has always been done, and it has worked for the Masters. These braces were planed flat from split flitches, to ensure integrity of the grain. (they are actually split from violin belly wood flitches, split in the Italian Alps in the 1980's....make your mouth water just holding it)
Putting the braces into some sort of tragic perspective, these cost me more than the price of three complete student budget guitars from my local music shop. What I'm doing just doesn't make cents [sic].