It's been touched on above, but when you consider questions like this you need to be concerned with the system parameters. When analyzing any system you may perform a static (or steady state) analysis, or you may perform a transient (or dynamic) analysis.

It is very appealing, intuitively, to consider systems as static and unchanging, as this reduces the complexity of the analysis. (This is why AC circuits and control systems are often studied in the frequency domain rather than the time domain. You can represent them using time-independent parameter values like frequency, phase angle and impedance and not include time in the calculations.)

So in the capacitor question, we can say that in a static system where nothing is changing with time, then current does not flow through an ideal capacitor.

However, in the real world, most systems of interest are dynamic (or they wouldn't do anything). In a dynamic system, current most certainly can flow through a capacitor.