The situation is as follows: You have a parallel plate capacitor, disconnected from a battery on both ends. We know the battery tries to maintain a constant potential difference between its two ends. The capacitor is uncharged. When then capacitor is connected to a cell, why do electrons start flowing? What is the need to create a field between the plates? EDIT: What is the need to maintain a potential difference between the plates?
Note what you say:
"We know the battery tries to maintain a constant potential difference between its two end"
The wires connecting the battery ends to the capacitor plates, by being conductive extend the function of the two ends
So the question refers to what makes a battery maintain a constant potential at its ends, which is answered in this link.
The battery acts like a 'pump' which creates a potential difference across its terminals. The potential difference causes electrons to accumulate on one place of capacitor. This repels electrons from the other plate into the battery hence completing the circuit. A battery acts jump of potential. You can consider it to be a circuit element such that there is an increase of potential across it from negative to positive terminal.