I had (and still have) some monkeypuzzle from a tree that was about 2' across.
The gum is exclusively in the bark, and the trick here is to get the bark off absolutely ASAP. Use gloves and avoid any other contact with clothes, as this gum will not come out. Once the bark is off (and it should actually peel off quite well), the wood itself is almost white, usually of very even grain and easy to turn.
However, sharp tools are essential, and since the tree has some silica in the wood, it will blunt your tools rapidly.
BTW, in its home countries (Chile and Argentina) this tree is now protected, because it was overharvested for timber framing. So much for no good for construction.