electromagnetism - Conservative $E$-field and Kirchoff rule in practice

In undergrad physics, when analyzing an LR circuit, it is often considered that Kirchoff rule holds. However, as far as I understand, Kirchoff rule only holds when E field is conservative (curl of E is 0, there is no change of flux). In a coil L, there is obviously a self-induced emf which makes closed loop integral E.dl non zero. My question is, does that mean we can only apply Faraday's law but not Kirchoff rule here.

Another question, even when there is no coil (only R's for example), in practice as we turn on the power supply, the current will create a B field inside the wire loop which in turn creates a change in magnetic flux. As a result, we again have a non-conservative E field here. Is it correct to say Kirchoff rule might never be applicable in practice? (at least to an absolute accuracy)

Answer

In a coil L, there is obviously a self-induced emf which makes closed loop integral E.dl non zero.

This is the EMF of a path encircling the core of the inductor, in roughly the path of one loop of the coil.

For lumped circuit analysis we aren't interested in this, but in the potential difference between one end of the coil and the other. This p.d. is indeed the EMF taken along the looping path of the coil. But as far as it interacts with the rest of the circuit, we can treat it as an electrostatic potential difference, which, when summed with the differences across the other elements in a loop of the circuit, will come to 0 according to Kirchoff's voltage law.

even when there is no coil (only R's for example), in practice as we turn on the power supply, the current will create a B field inside the wire loop which in turn creates a change in magnetic flux. As a result, we again have a non-conservative E field here. Is it correct to say Kirchoff rule might never be applicable in practice?

This is indeed the issue when we talk about the lumped circuit approximation. The magnetic flux in the loop formed by the circuit elements is not accounted for in lumped circuit analysis.

And indeed the lumped circuit approximation is an approximation; it is not absolutely correct.

But often the errors introduced by this approximation are much smaller than other errors in our analysis (for example due to not knowing the characteristics of the components accurately, not knowing the temperature of the components accurately, etc.) So in practice, Kirchoff's voltage law is usefully applicable in a great many real circuits.