(That photo of the chap sitting on the elephant is not Bell.)

As written above, Bell liked the small fast calibers for deer. .22 Hipower was a favourite, but the .220 Swift was his primary one after he got his Winchester. But he also had used the 7x57 and 6.5x54 on red deer as well.

I have not used the .220 Swift on deer, although I know people who use the .22-250 and they say it does indeed kill like lightening. I have used the .222 on red deer (at bush hunting ranges) and frankly they all died. I do not understand the UK energy minimums, they sound like something a bureaucrat who knew nothing about the subject would dream up to measure "killing power". You might as well measure how much smoke comes out of the barrel. (I have killed red deer with the .44-40 and black powder loads. An energy value of about 400ftpnds I think. Every bullet passed completely though and they all died within twenty feet.)
For example the .222 and .223 have been used in New Zealand on red deer both big and small by both recreational shooters and professional for decades. They would not be used if they were not effective.

And a .222 will kill in the right place as well as a .243 or a .270, whereas a bullet from any of them in the wrong place will not work at all.
I think I just paraphrased Bell

Bell initially gave up on his Fraser 6.5 Mannlicher Schoenauer carbine because the ammunition was unreliable. This was common with most British sporting ammunition of the day. Misfires were common, Cases rupturing and blowing gas into the face of the shooter were a constant hazard. The most reliable cartridges of the time were military - the British .303 and the Mauser 7mm as made by DMW, and this is what he used for elephant.
I think cartridge reliability has a lot to do with his choices in the early days, and he says as much in one of his books - he is quite scathing about British sporting ammunition.

But what many people don't realise nowadays since he is such a legend for using the 7x57 on elephant, is the fact that by the outbreak of WW1, and then his safaris after the war, he had settled on the .318 Westley Richards with the 250 grain bullet as the best cartridge for his style of elephant hunting. He used several .318's also.
From about 1912 onwards he was a .318 Express shooter, as far as African game go. This puts him in line with many other hunters in the British empire.