The charging time of capacitor depends on its energy absorption, which can be used directly to get the drift velocity. Looks like I answered my own question.
I was caught up in the notion that electrons never actually enter or leave a wire. I'm not sure how i reached that idea because...
I'm having trouble understanding where the charging time in a capacitor actually comes from. Is it possible to derive the \tau of a capacitive circuit from the drift velocity of the electrons? Are charges literally moving from the conductors onto the metal plates of the capacitor?