Does a weak acid undergo it's weak ionization upon the addition of any kind of base?
A weak acid does not (contrary to what you show in your equation) predominantly ionize before the base is added. But keep in mind that proton transfer reactions (at least those between oxygen and/or nitrogen atoms) occur almost instantaneously in both directions, so the equilibrium you illustrated is established immediately (for a weak acid, with the big arrow pointing left). So when (strong) base is added, H3O+ is consumed immediately, then more acid is ionized to re establish the equilibrium--and this goes on as you add more base until you have consumed almost all of the weak acid. These reactions are controlled by the relative energies of the reactants and products, and not by reaction rates.