The Mini gearbox was never designed to take the forces and stress involved when you get to the higher ratio diffs.
I think you are correct Ethel that you would expect the torque effect to be less, but I think its the fact that with the torque on tap it struggles to pull that ratio and instead of turning things quickly it breaks something...?
The mechanical leverage makes higher ratios more difficult to turn. So you need more torque to accelerate at the same rate as a lower diff....if you see what I mean...its harder and slower to accelerate in 4th than 3rd, and the final drive is like all the other gears in any gearbox, just a constant ratio.
That's why you get poor acceleration with diff ratios higher than 3:1 and they are more prone to breaking, big wheels make it worse.
With low ratio diffs the leverage ratio is there to help things accelerate quicker by turning easier, hence more fun, acceleration and wheelspin potential.
The Cooper S diff ratios were an option with both standard and close ratio boxes...
970 and 1071 ; 3.76 standard with the options of 3.93 4.13 and 4.27
1275 ; 3.44 standard with the options of 3.76 3.93 and 4.13