Electrolysis occurs when a non-spontaneous redox reaction occurs when electrical energy is put into it. An electrochemical cell functions by using two redox reactions to facilitate movement of electrons. So for an alkaline battery, the zinc anode will oxidize and release electrons, which will flow through the circuit to reach the manganese ($\ce{MnO2}$) cathode, which will be reduced, allowing more electrons to flow through the circuit without buildup of electric charge. I assume here that both reactions are spontaneous, and the electron buildup in the anode prevents further zinc oxidation when the circuit is incomplete. $$\ce{Zn(s) + 2OH- (aq) \rightarrow ZnO(s) + H2O(l) +2e-} [E^\theta= 1.28V]$$ $$\ce{2MnO2(s) + H2O(l) + 2e- \rightarrow Mn2O3(s) +2OH- (aq)}\ [E^\theta= 0.15V]$$
If the alkaline battery is linked to an electrolytic cell, won't it just be like two electrochemical cells working in conjunction to each other? The zinc anode will be linked to the cathode of the electrolytic cell, and then the ensuing redox reaction will (probably) be spontaneous. So the "electrical energy" put into the electrolytic cell will be instead chemical energy?
Also, seeing that the $\ce{MnO2}$ cathode has a relatively small $E^\theta$ value, won't it be hard to trigger oxidation of the electrolytic anode? When connected to the electrolytic cell, do the cathode and anode of the electrochemical cell now work totally independently? If that is the case, the voltage of the battery should not be considered, but instead the electrode potentials of the anode and cathode, but that is usually not the case. What am I misunderstanding about this?
edit: I'm asking about the mechanism when an electrochemical cell is linked to an electrolytic cell, not about the reactions taking place in the electrochemical cell itself, so the question suggested doesn't quite fit the bill.
edit 2: Added a picture to better show my doubts. If the galvanic cell is linked to an electrolytic one as shown here, the galvanic anode and the electrolytic cathode are linked and seemingly acts as a separate cell. The same thing can be said about the other two electrodes. If this is the case, doesn't the galvanic anode and cathode each act independently of one another, and therefore the electrode potential of the individual half cell must be considered? If they are actually linked, how?