Wood is a superb material for aeroplanes - how else did the Mosquito do so well, and for that matter the Minimax?
There are however some problems that need to be overcome, that have always been a problem.
(1) There are many many sorts of trees in the world, and each types wood has subtly different engineering properties. So, either you need an incredible amount of engineering data, and to some extent to keep generating new data with different samples - or you restrict yourself to a few very expensive woods.
(2) The manufacturing is very hard to automate.
(3) People with the right skill-sets are increasingly rare.
(1) and (2) are both solveable, but it'll always remain fairly labour intensive. However, the fact is that none of these problems exist for aluminium alloys, nor even to a fair extent for modern composites - and they do add cost.
G