"If you _didn't_ know squared amplitudes corresponded to probability of experiencing a state, would you still be able to derive "nonunitary operator -> superpowers?""

Scott looks at a specific class of models where you assume that your state is a vector of amplitudes, and then you use a p-norm to get the corresponding probabilities. If you demand that the time evolutions be norm-preserving then you're stuck with permutations. If you allow non-norm-preserving time evolution, then you have to readjust the normalization before calculating the probabilities in order to make them add up to 1. This readjustment of the norm is nonlinear. It results in superpowers. The paper in pdf and other formats is here.