well the thinking at the time was when firing at the massed assaults in WWI they would be sometimes 2500-3000 yds away so the cut off was used to fire one shell at a time keeping the mag full for when they got close. I guess being that the Europeans were fighting for 3 years the Americans figured to copy the enfield
Logistics and natural conservatism in the brass was the reason. Ammo had to be be brought out to the front by horses when the lee enfield, krag, and 1903 were designed. This, the rationale was troops would waste ammo that could not be easily replaced without a cutoff. The 1903 retained it after wwi probably because of parade maneuvers through the 03a3.
Having a magazine cut-off on the Krag made sense, on the M1903, it did not. THe Germans came to this conclusion when they adopted stripper-clip loading in the Mauser, which do not have this feature.
With the Krag, each round of ammunition was stored in a individual loop on the cartridge belt, and had to be individually removed from the cartridge belt and placed in the magazine, so the average time required to load and fire 5 rounds was the same whether you fire single shot (one round, chamber, fire, repeat four more times), or fired five rounds from the magazine (one round in the magazine, repeat four times, then shoot five times). But, if you used the cut-off in the "OFF" position, you could maintain a constant rate of fire equal to how long it took to pull one round from the belt, load and fire, but you got to keep the five rounds in the magazine as reserve in case you needed a quick burst of five rounds.
Once you adopted the stripper clip, the magazine cut-off becomes pointless. Now, all ammunition is going to be issued and stored in the cartridge belt in stripper clips. So, if you were going to shoot one round at a time, keeping the five rounds in the magazine "in reserve", you would have to remove a stripper clip from the belt, remove one round from the stripper clip, load it into the rifle, do something with the partially loaded stripper clip (put it in your pocket, or replace it in the cartridge belt, place it on the trench parapet, etc), then shoot. Now, you also have a partially loaded stripper clip, that can be potentially lost.
With a the speed that a stripper clip can be loaded into a magazine, and the fact that all ammunition is pre-packaged in stripper clips, it just make no sense to have a cut-off. The stripper clip actually makes loading single rounds more difficult, and increases the chances of lost or wasted ammunition if ammunition is removed from the stripper clip.
The M1903A3 kept the cut-off because it easier to keep it on than to remove it.
Similarly, the magazine cut-off in the MLE and LEC made sense, on the SMLEs and CLMLEs it should have been removed, it was pointless.