You are correct that the C cannot charge up due to the low resistance of the secondary winding that would always discharge it. This is a mistake on my part.
However any DC bias currents flowing in or out of the + or - amplifier inputs have no route to 0V. All of the protection diodes would be normally be off, apart from leakage currents, at the normal voltage levels at the + and - pins of the differential amplifier around about 0V.
The capacitor will provide some filtering of the transformer output signal and reduce the bandwidth at some point.