I think you sort of need the thin sandwiched material to get a good sound from it. There needs to be a "membrane" for it to be a drum. The more solid your materials most likely the less sound/projection you'll get. You want a balance between too hard which is echo-ey to too soft which will break and produce a light sound.
You might be onto something. I have a $50 thin plywood cajon (made by a relatively local joint called DrumBox Percussion) and a $200 Toca cajon which I think is made of oak. The Toca looks beautiful and has a crisper sound but feels too stiff. The DrumBox does sound a bit cheaper but has more oomph in the bass sound and is more playable, and also more light and portable.