We should be able to treat DC as an AC case with $\omega=0$. But then $\hat{Z}_C$ is arbitrarily large, and in an $RC$ circuit we should have zero voltage across the resistor. But in an $RC$ circuit the voltage across the resistor is nonzero for some time. Why is there a disagreement?

The key to resolving this is to realize that for a DC circuit, in steady state, there is no current across a capacitor (note: zero current, not zero voltage). The capacitor basically acts like an open circuit. For the case of DC battery, capacitor, and resistor in series, at infinite time, there is no current and no voltage drop across the resistor (i.e., both sides of the resistor are at the same potential), so the voltage drop across the capacitor is the same as the drop across the battery. This is consistent with the $\omega\rightarrow 0$ limit of AC Ohm's Law: $\hat{Z}_C$ is infinite, and therefore $\hat{I} = \hat{V}/\hat{Z}_C$ is zero.

Now, there are transient solutions to the DE's set up using the Loop Rule, as we saw a few lectures ago: for the short period of time while the capacitor is charging up, or discharging, there is a varying current (and voltage) across the capacitor. However the AC limit corresponds to the steady state situation, after transients have died away; for an $RC$ circuit, potential across the capacitor is constant and current is zero after the transients have gone away.