A capacitor shouldn't allow electron flow between its (dielectric separated) plates. The charges should build up on the plates, then exit through the terminals when discharging. If charges cross the dielectric, then you have leakage.

Hm. Apparently my Phys101-level understanding of electronics has caught up to me.

Caps don’t allow electrons through, do they? They resist DC, they only conduct AC?

That is correct. The dielectric ‘conducts’ an electric field, which attracts and repels electrons on the opposing plates.