Yes, my dyslexic typing again. I meant to say: They would have gone with cock on opening.Seem like you might have some contradictory/incorrect typo going on there. Happens to me all the time. . .
As far as rapid fire or rate of fire for troops goes:
The Krag and the 03 had a magazine cut off device as well as the original Pre * SMLE. Sergeants controlled their squad's and platoon's rates of fire much more closely than they do today. The magazine blocks were designed to aid that when the soldiers were engaged in sustained fire although in practice, that really didn't work out too well to my knowledge.
One of the key modifications to the British Enfield rifles after the Boar Wars was the addition of a clip guide and ammunition issued in 5 round clips. This enabled the Tommys to fire rapidly after their experiences with the German Mausers of the Boars.
The 1903 was designed from the outset to have a clip guide after the US Army's experiences with the German Mausers of the Spanish Army in Cuba circa 1898. This allowed US soldiers to match the rate of fire of the Spanish Mausers. Much is made of the US Ordnance Department paying royalties to Mauser for patent infringements of the 03. While not normally mentioned, one of those infringements was the clip guide milled into the receiver's bridge.