Inductor with AC source
You have missed one thing: The induced voltage is the result of a change in the current. The current must be changing, otherwise no induced voltage exists. The current changes in a way that the voltage which is caused by the induction between the terminals of the inductor is just equal with the fed voltage (assuming resistance=0)

You seemingly think the induced voltage is there at first and then "no voltage difference, no current can be born" That's reversing the cause and the consequence. As well one could think a freely falling object cannot fall because the air resistance has grown as high as the gravity. The error is the same. Air resistance needs the motion to exist.

The induction law in ideal coils (=no resistance, no capacitance, no radiation as radiowaves) has the following quantitative form: The current changes at rate = fed voltage divided by the inductance. That's the formula for the time derivative of the current.

If you connect A volts constant DC to a coil which is ideal and has inductance = B Henrys the current grows infinitely at rate A/B. If A=1V and B=1H, the current grows one ampere per second.