The biggest reason is mass-manufacturability. It's so cheap and easy to stamp aluminum enclosures once you've done the tooling, while wood requires milling each piece. But if you're doing something one-off or for aesthetic reasons, the reproduction cost shouldn't be a concern.
For the other reasons:
Thermal - you're right.
RF - chicken wire would allow high frequencies through. If the wavelength of the EM wave is smaller than the opening in the metal, it can get through. Aluminum foil doesn't, AFAIK, have a plastic layer. It should be good enough. But it tears easily, so you have to be pretty vigilant to get good coverage. Also, you need some spring mesh or something around any flanges or lids etc to ensure the electrical connection carries around the corner. Note contact should be continuous all the way around any opening. In theory, the thickness of the metal shouldn't impact the shielding (as long as the metal doesn't tear) but I don't know enough RF engineering to really assert this.
Strength - if your box sits on the floor, wood should do fine. If it mounts in a rack, you'll have to decide what to do about the mounting flange. The shear strength of wood is not as good as metal, so you might have to fabricate the flange out of metal, and use a lot of bolts to connect it to the wood. Wood is heavier for a given volume so the flange will have to carry more load.
Earth - I'm not too sure about this one. I guess lightening strikes are the biggest reason you need to worry about this, and I'm not sure how much protection the metal case would give you anyway in that situation. Anyway, your RF cage needs to be connected to ground so maybe you'll just end up with some charred aluminum foil to clean up. If someone pokes something metal through the case and makes contact with the power supply, thicker metal on the RF cage would help.