Believe it or not material for buttons was in extremely high demand and southern armies/depots started to rely on anything they could utilize, so flat buttons are very common. Many buttons were pre-war or imported via blockade, though there were a spattering of local manufacturers in the South. Wood buttons were in ready use in the depots mid to late '64 on. Buttons were even cast in the field by soldiers themselves, out of any material they might be able to melt, lead, brass, copper etc. I have a couple of field cast Maryland buttons.
1864 copper and brass were in very short supply and were desperately needed for percussion cap production, so unless the buttons came in through the blockade they were taken from other uniforms (thus the proliferation of US General Service buttons, field cast, wood and yes flat).
Keep in mind also that uniforms were worn after the war, particularly the jackets and an ordinance was passed that did not allow for the display of any insignia of the Confederacy. Soldiers who needed to wear the jackets through necessity or pride would have to strip the buttons off the jackets and replace with an unadorned piece, this has greatly confused the collecting community when studying the extant jackets. Once the ordinance no longer made sense, buttons would be sewn back on, left the same or sewn on later by a collector.