Similar designs have been around for a long time - in fact, treadle-powered bikes pre-date James Starley's perfection of the modern chain drive in the late 19th Century. The fact that they were almost immediately and entirely replaced by chain-driven bicycles probably tells us all we need to know.

At guess, I'd be surprised if this system could rival a chain drive for efficiency: a well-maintained bicycle is an extremely efficient machine and a single-geared clean chain under ideal conditions can approach 99% efficient power transfer, while even in normal use (add dirt, gears etc) efficiency is around 90%.