Can anyone explain why a B-H curve with Hysteresis doesn't result in a highly non-linear input-output transfer function for a transformer?
"the H field generated is proportional to the current in the primary" ... no, it's proportional to the difference between primary and secondary current (latter scaled by turns ratio). 

So the nonlinearity only applies to the magnetising current. Now, if you drive the transformer from a finite source impedance, that nonlinear current translates to a nonlinearity in the voltage across the transformer itself.

Assume 10% nonlinearity from the region of the B-H curve we actually use, and the magnetising current is 10% of the total, the nonlinearity in the primary current is 1%.

Furthermore, assume the source impedance is 10% of the load impedance (scaled by N^2) and primary impedance in parallel, so that 10% of the driving voltage is developed across it by the total primary current: that 1% current nonlinearity results in 0.1% nonlinearity on the secondary voltage.

So a 10% B-H nonlinearity in this example results in 0.1% nonlinearity in the transfer function. And to reduce it further you can reduce the source impedance of whatever drives the transformer.