Are you using quartersawn White Oak? In our area, Red Oak is significantly
more expensive than White (unless quartersawn).
I play with using contrasting hardwoods in some projects and have used Red
and White Oak together but you need to use them the same way in various
parts of the project (example legs, drawer fronts/facings, doors, tops, etc
should not share woods). Oak ply would probably be a better choice than
birch for the hidden areas that will not take a lot of direct ambient light.
Don't overlook Ash for some of the minor parts. Ash can do a good Oak
imitation especially Red Oak. You will have to experiment, on some scrap,
with stains to make sure everything can come together, especially with the
darker colors. Also, Ash can blotch so you might want to condition the
project with spririts or a commercial pre-stain conditioner. I have seen
several pieces of the dark red mission furniture that is actually made from