Why do we start the engine on a PW before introducing fresh water when flushing?

Historically because if you force water into the engine without it running, water can get into the carb and cylinders causing something called hydro-lock. 

Slightly different with the new closed loop cooling systems but its still possible to flood parts of the engine and from a corrosion point of view flushing through the exhaust etc has enormous value. Engine running = water traveling in the direction it was designed to travel through the PWC. 

Hydro-lock = Once a cylinder fills with water, the compression stroke jams or 'locks' the engine (because water doesn't compress unlike air) and the engine cannot move.