How can $I_c$ be zero if $V_{CC}>0$? Doesn't $V=IR$ imply the current is only zero when the voltage is?

No, in the switch, when $V_1$ is small, there is zero $I_b$ and zero $I_c$. In this case, you can think of the transistor as acting like an open circuit that does not allow current to flow (inside it, the depletion region acts like an open capacitor). If there is no current through the resistor, there is no voltage drop across $R_c$, and the whole branch at the transistor collector up to the supply is at $V_{cc}$.