1. The Enfield bolt cocks on closing, which makes rapid fire unhandy, to say the least. I too own a No. I Mk 3, and a genuine Jungle Carbine and they are not fast, even compared to other bolt-action rifles.
2. And most importantly, our luckless future Star Fleet Chief Engineer would have fallen straight down and out of the way of the other shots on the first hit by a .303. There is no flingin' way any human being would stay standing after one hit. Whatever hit him was full-auto and pistol caliber, to get five rounds into him before he collapsed.
I should know a bit about these things, since I spent about 17 months shooting and hitting other people and then getting hit myself.
The Lee–Enfield rifle was derived from the earlier Lee–Metford, a mechanically similar black-powder rifle, which combined James Paris Lee's rear-locking bolt system that had a barrel featuring rifling designed by William Ellis Metford. The Lee action cocked the striker on the closing stroke of the bolt, making the initial opening much faster and easier compared to the "cock on opening" (i.e., the firing pin cocks upon opening the bolt) of the Mauser Gewehr 98 design. The bolt has a relatively short bolt throw and features rear-mounted lugs and the bolt operating handle places the bolt knob just rearwards of the trigger at a favourable ergonomic position close to the operator's hand. The action features helical locking surfaces (the technical term is interrupted threading). This means that final head space is not achieved until the bolt handle is turned down all the way. The British probably used helical locking lugs to allow for chambering imperfect or dirty ammunition and that the closing cam action is distributed over the entire mating faces of both bolt and receiver lugs. This is one reason the bolt closure feels smooth. The rifle was also equipped with a detachable sheet-steel, 10-round, double-column magazine, a very modern development in its day. Originally, the concept of a detachable magazine was opposed in some British Army circles, as some feared that the private soldier might be likely to lose the magazine during field campaigns. Early models of the Lee–Metford and Lee–Enfield even used a short length of chain to secure the magazine to the rifle.[10] To further facilitate rapid aimed fire the rifle can be cycled by most riflemen without loss of sight picture.These design features facilitate rapid cycling and fire compared to other bolt-action designs like the Mauser.[5] The Lee bolt-action and 10-round magazine capacity enabled a well-trained rifleman to perform the "mad minute" firing 20 to 30 aimed rounds in 60 seconds, making the Lee–Enfield the fastest military bolt-action rifle of the day. The current world record for aimed bolt-action fire was set in 1914 by a musketry instructor in the British Army—Sergeant Instructor Snoxall—who placed 38 rounds into a 12-inch-wide (300 mm) target at 300 yards (270 m) in one minute.[11] Some straight-pull bolt-action rifles were thought faster, but lacked the simplicity, reliability, and generous magazine capacity of the Lee–Enfield. Several First World War accounts tell of British troops repelling German attackers who subsequently reported that they had encountered machine guns, when in fact it was simply a group of well-trained riflemen armed with SMLE Mk III rifles.[12][13]
It does indeed cock on closing (I also have one, and can confirm), but Sergeant Instructor Snoxall doesn't seem to have found it "unhandy". My only point of comparison for military bolt-action rifles is the Mosin-Nagant. I can manipulate it faster than a MN ...
As for getting hit in the leg ... Men have used the .303 to kill elephants.
I have found my No. 4 particularly quick and easy to fire. May just be me.