There were several reasons that Bell used small bores in favour of the more conventional big bores:

He carried his own rifle. Bell put on many miles a day in hot conditions and a heavier load would have prevented him with putting on those extra miles.

Many of the solids used in the big bores were not up to the job. The jackets split and pealed. Penetration was not a sure thing.

As he got into the herds he would shoot a large volume of ammunition taking large numbers of elephant.
This would not be possible with a big bore double.

He stated that the 7 mm Mauser, aka, the .275 Rigby, with DWM ammunition was the only cartridge that never failed to go bang, always penetrated, or never suffered from split necks ( poor annealing was apparently a big problem in those days). Every other calibre and brand had given him problems at one time or another.

He often has a gun bearer with him who carried along a big double for emergencies. I believe it was a .450 Nitro Express. He mentioned this in one of his books. They are all a great read.

There will only ever be one W.D.M. Bell. They broke the mould.