The main purpose of whiteboard is to decrease the liquidity of job search for engineers, thus depressing wages. After a few years of doing design docs and copying protobuf fields, the median engineer at a top company would need a month of prep to be able to pass whiteboard interviews again. Much more effective than secret no-poaching agreements.
A less intriguing but probably more accurate reason is that finding good people is hard, nobody really knows how to do it, whiteboard interviews provide a rough filter, and they're what everybody else does, so we're gonna do it....