Total changing flux in Faraday's law of induction
The conductor is a loop of wire hence it has self inductance. Do not lose sight of that because, ultimately, this means the net flux can never be cancelled to zero.

So, imagine the loop were opened and the voltage measured. With a steady rising external field, the voltage induced is constant because of the rate of change of flux being constant.

Now with the loop closed and taking into account the loops self inductance, the current flow is a ramp (because of the constant voltage and inductance ignoring DC resistance) that produces an opposing ramping magnetic field inside the coil but an increaded magnetic field outside the coil and local to it: -



The two fields cannot cancel else there would be no induced voltage to create the counteracting magnetic field. The net effect is no change to the average flux density of the external B field.