I'm going out on a limb here, but I will suggest that John Kingsley-Heath recommended the.243 because most of his clients could better put a .243 bullet in the right place than could be done with a hard-kicking magnum.
That's definitely part of his recommendation--but JKH also preferred the .243 for his own leopard hunting. Since he didn't handload (it was illegal in several African countries back then) he was using Winchester's 100-grain Power Point load--which was the most commonly available factory ammo in Africa in those days.
If I recall correctly, he also used his .243 and that factory load when he hunted Ethiopia extensively when looking for a black "panther," the melanistic version of leopards, which was apparently more common in Ethiopia. (The spots are still barely there, but the angle of the light has to be just right to see them.) So while the light recoil was part of his recommendation, he also believed in the .243 for the other reason stated in my post: less damage to the hide than bigger cartridges. His pre-'64 Model 70 was one of his favorite rifles, and many of his clients used it as well.
Again, it's always interesting to read so many firm opinions on appropriate rounds for animals the poster has never even seen, much less hunted. Have hunted with a bunch of African PHs over the decades, and they'll frequently disagree--but often the most experienced have different opinions even on rounds for much larger game.
One example is the PH I've hunted with most, the now-retired Kevin Thomas, who was born and raised in what was then Rhodesia. He killed his first charging Cape buffalo at 17, on his first job out of high school, when working as a "game scout" for the Rhodesian game department. This mostly involved guiding sight-seeing clients in various parks, while carrying a .375 H&H just in case some animal objected to being looked at.
One of his jobs a few years later was as the wildlife "manager" on a huge Rhodesian cattle ranch--which basically meant culling wild game to leave more forage for cattle, since there was basically no safari industry in Rhodesia then, as beef cattle were more valuable. He used various cartridges and bullets--but one night while we were sitting around the campfire I asked him what he considered the absolute minimum cartridge for hunting buffalo.
He thought about it silently for a couple of minutes before answering, when he said, "The .30-06." I raised my eyebrows and asked why. It turned out he'd used a .30-06 with handloads featuring the original lathe-turned 180-grained Nosler Partition, to cull over 500 buffalo. This was not at night with head shots, but during daylight with the native ranch workers pushed herds past him. He shot every buffalo that offered a decent shot--calves, cows, bulls of all sizes--and said he never had a problem, even with frontal shots on mature bulls. He also preferred the .375 H&H as his back-up rifle on buffalo when he became a PH a few years later, mentioning that he often had to follow-up and finish bulls that his clients wounded with much larger cartridges.
