Applicability of the concept of voltage in electrodynamic circuits

Yes, it is possible to use the concept of voltage and related tools in AC circuits. The fact that the contour integral of $\vec E$ isn't zero isn't a problem. It would only be a problem if you first, "gradient" equation were right in this context. However, it's not. The correct equation needed for more general electromagnetic setups is $$ \vec E = -\nabla \Phi - \frac{\partial \vec A}{\partial t} $$ The identity for $\vec E$ implied by this definition isn't simply ${\rm curl}\,\vec E=0$ but the full Faraday's law Maxwell's equation you wrote down $$ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} $$ because $\vec B = \nabla\times \vec A$. The equations above are fully consistent with one another and compatible with the fact that $\Phi$, the vector potential, is a well-defined field even in general situations involving time-dependent fields and magnetic fields.

There is an ambiguity in the choice of $\Phi,\vec A$, the gauge invariance, but useful conventions may be assumed to fix this ambiguity in the case of AC circuits. When it's done, the non-magnetic parts of the circuits work just like before or in DC circuits. The voltage may be calculated as the differences of $\Phi$ just like in the DC case.

For coils, one has to appreciate that the integrated electric field isn't the only relevant contribution. Instead, one finds the electromotive force, EMF, a modified sibling of "ordinary" voltage which has to be added to coils (and batteries) for the total sums of voltages over closed loops to vanish.

One may also study harmonic time-dependence in which all quantities depend on time as $\cos\omega t$ which is usually complexified to $\exp(i\omega t)$.