This may be a very basic question but my misunderstanding is due to the difference between how physics explains the charging of capacitors and how we analyze circuits in circuit analysis.
According to physics, when a battery is connected to an uncharged capacitor, electrons flow from one plate to the other creating an electric field, and hence a potential difference across the capacitor. This potential difference starts at 0 and increases until it reaches the voltage of the battery.
However, this means that for the time the capacitor is charging, the potential difference across it is different from the potential difference of the battery. According to circuit analysis (or at least the lumped-element model) this connection is not valid. The definition of 2 elements being parallel is that they have the same voltage.
The only solution I can think of right now is that it is indeed valid to have this connection however in real-life circuits there is no ideal short circuits so if we connect a battery to a capacitor, the wires have a bit of resistance and so we can model them as resistors in the circuit. So when the battery is connected, the resistance will have all the battery's voltage at first while the capacitor will have 0 volts. The current that flows is due to the capacitor charging (and will be very high because the resistance of the wire is very low). As the capacitor is charging, its voltage increases while the voltage of the resistor decreases until the capacitor takes on all the battery's voltage causing the current to be zero and so the resistor's voltage becomes 0.
And that's why analyzing the step response of a capacitor, there's always a resistor (usually with a high resistance, on the order of 10 kilo Ohms) connected in series with the capacitor.
Is this really the case? Or am I missing something?