You're right on the money with the rapid fire, Rule303. British doctrine at the time stressed high-rate, aimed fire (Battle of Mons). They found cock-on-closing led to a higher rate of fire, for several reasons, such as extraction. Remember, this is for men who had little or no experience with shooting, especially bolt actions, prior to going into the army, so they had no preconceived preferences to overcome, no habits to unlearn.

(Almost all my rifles are cock-on-close; I'm uncomfortable using the other system, which feels unnatural to me.)

Cock-on-closing wasn't the only element of rapid-fire design. They also looked at such things as bolt handle placement in relation to the trigger. Few people seem to understand that the curious dogleg shape of the P13/P14 Enfield bolt handle was as a result of rapid-fire testing with the prototypes. I'm very fond of the shape on mine, but others go to great lengths to reshape them into something bland (Mauser-like.)

Off-topic bit of trivia: The British Army wasn't alone in their insistence on a high rate of aimed fire. The Royal Navy was equally enamoured with the practice. Some have made the case that their desire to have quick access to ammunition during battle led to the stockpiling of cordite outside of masgazines, and failing to keep blast doors closed to speed reloading. The result may have had much to do with the loss of so many battle cruisers to magazine explosions.