Nice post - thanks for taking the trouble to write this, and make figures!

The NBS has other counter-intuitive properties, such as the that extra solutions can make some players worse off.

Why would that be counter-intuitive? Saying that is counter-intuitive implies that you should choose the max-min solution (the solution with the max over players P of(min(utility(P)) ).

Hence the KSBS replaces the independence requirement with "if one adds an extra option that is not better than either player's best option, this makes no player worse off".

You need to explain what "better" means in this context. Better for both players? With a sum or product that is higher? I can't think of any meaning that makes sense.

Another way to formulate the KSBS is by renormalising the utilities so that d=(0,0), u=(1,1), and then choosing the egalitarian solution:

What defines "the egalitarian solution"?