Let's look at your first statement:
A thermodynamic transformation that has a path (in its state space) that lies on the surface of its equation of state (e.g., $PV=NkT$) is always reversible
I don't think this is right, but there may be some implicit qualifiers in your statement that I'm missing.
Here's an example to illustrate why. Consider a system consisting of a thermally insulated container with a partition in the middle dividing it into compartments $A$ and $B$. Suppose that we fill each side of the container with an equal number of molecules of a certain monatomic ideal gas. Let the temperature of the gas in compartment $A$ be $T_A$ and the temperature of the gas in compartment $B$ be $T_B$. Suppose further that the partition is designed to very very slowly allow heat transfer between the compartments.
If $T_B>T_A$, then heat will irreversibly transfer from compartment $B$ to compartment $A$ until the gases come into thermal equilibrium. During this whole process, each compartment has a well-defined value for each of its state variables since the whole process was slow ("infinitely" slow in an idealized setup). However, the process was not reversible because heat spontaneously flowed from a hotter body to a colder body. You can also show that the sum of the entropies of the subsystems $A$ and $B$ increased during the process to convince yourself of this.
So we have found a process for which the thermodynamic evolution of each subsystem can be described by a continuous curve in its state space, but the process is irreversible.