Why does a IRFZ44N turn fully on on my battery discharger while a IRFZ46N doesn't?
You should not have measured the battery voltage on the gate.

The datasheet for irfz46N states:

VGS(th) Gate Threshold Voltage 2.0 ––– 4.0 V VDS = VGS, ID = 250μA

1.2V would not do much on this device. You can suspect issues with the gate driving circuit.

This is the UI characteristic of IRFZ46N. Parameter is Vgs.