The current in the field or exciter coil creates an ac magnetic field that induces a voltage, not a current, in series with the pickup coil. This follows from Maxwell's equation or the law of magnetic induction. Current flows if there is a load on the coil, such as the coil capacitance or the loading caused by eddy currents in the cores, etc. (A so called current transformer is a tightly coupled transformer driven from a high impedance so that the current in the secondary is related to that in the primary by the turns ratio. The very loosely coupled situation here does not behave that way.)
You describe one way to make the current through the field coil independent of frequency: make the inductive reactance low compared to the dc resistance of the coil across the whole useful frequency range. A better way is to drive the coil with a current source, that is, a circuit with an output impedance much higher than the impedance of the coil at any useful frequency.