Just for the Comment;
Can I understand your statement quoted below is same as Box 1 below?
The reasoning goes that because entropy is a state function, this relation holds even for irreversible processes because we can imagine that there is a reversible path between the two states.
If so, all states shall be connectable by path, but it is not necessary that the path is reversible. In other words, as long as the path exists, everything is "ideal" in the sense of Box1.
Box1:
- To define entropy (exchanged entropy; $S_e$), for any two states A and B, there is at least one "ideal" path between A and B.
- If $c_1$,$c_2$ are the "ideal" path between states A and B, then the following holds;
$$\int_{c_1} d{S_e} =\int_{c_2} d{S_e}$$
We call it a "quasi-static process" in which U, V ,N ,and ... are fixed at any stage of the reaction. This quasi-static process is used as a method to realize a path that can be considered as a curve in the (upper half of) Euclidean space.
- When we say that "entropy is a state quantity," that "entropy" is the exchange entropy.
- If the path were irreversible, then a new term, called generated entropy ($S_g$), will be generated. Clausius' inequality is not an equality if this generative entropy exists. Clausius' inequality is not an equality if this generative entropy exists.
Indeed as described here, $\delta Q $ is path-dependent. But, U, V, N and T are state quantities and your following equation is path-independent
$${dS}_{e}=dU/T+P/T dV +\mu/T dN$$
and if we define the new state quantity $\delta Q_{rev}$, the $\delta Q_{rev}$
$$\delta Q_{rev}=dU+P dV +\mu dN$$
then, the $\delta Q_{rev}/T$ is a closed form and having a potential.
So once we admit that $dS_e$ can be written as the above equation, whether the curve is of a reversible process or not, as long as we keep track of U, V, N, and T from time to time, we can "recover" the function $S_e$ even if we don't know the so-called $\delta Q$ itself, IMO.
But, it would be simpler to make it an axiom that there is something called "exchange entropy".