Chemistry - Linking an electrochemical cell to an electrolytic cell
For an electrolytic cell, a galvanic cell is nothing else but a power source.
For a galvanic cell, an electrolytic cell is nothing else but a powered electronic circuit.
The galvanic cell anode/cathode does not form with the electrolytic cell cathode/anode a respective separate cell, as there is no ionic flow between them. Similarly for a simpler case, 2 half cells do not form a cell if there is no ionic connection between them.
The reason for it is simple, based on the electrostatic neutrality requirement and enormous forces between unbalanced charges. Imagine you managed to push the current 1 A for 20 s through the wire, without being balanced by ionic motion. The comparable unbalanced charge causes in atmosphere a lightning several kilometres long.
You seem to be mixing a couple of things. One important clarification regarding
Also, seeing that the MnO2 cathode has a relatively small E$^o$ 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?
The listed electrode potentials are valid only in relatively dilute solutions under very specific conditions. Don't apply the half cells on a dry cell or an alkaline cell. In short, those numbers are not meaningful for an alkaline battery or a carbon-zinc dry cell. Those commercial cell compositions is just a magical mixture, which is more art than science. Battery technology is not that trivial. Think why there are not many types of batteries despite 300 years of research?
If you consider a galvanic cell such as your alkaline battery as a simple water pump, then a lot of your confusions will go away. Its only job is to "circulate" the electrons. When you connect a galvanic cell to an electrolytic cell, the anode and cathodes can no longer be considered independent. If the galvanic cell pumps out one electron, the electron has to travel the cathode, anode and the external circuit of the electrolytic cell. In that process, something in the electrolytic cell has to chemically decompose!