It's because the phone design needs the battery to help stabilize the voltage under load. As we know, digital devices can nearly instantaneously change the amount of current they consume and thus require layers of energy storage to accommodate the transient currents quickly. However, the quickly changing current consumption doesn't just happen briefly. It sometimes continues to ramp for milliseconds. Thus generally every component in the power supply network of a design serves some stabilization and filtering role, including the batteries.

It appears that in this case, as the original battery aged, its internal apparent resistance (ESR) increased beyond the original design expectations, to a point where the phone won't work when plugged in to a charging cable because despite the charging cable most likely being able to deliver sufficient power at DC, it had too much impedance to supply it quickly enough. When current is demanded from a source that has too high impedance to supply it, the voltage drops. This will result in significant voltage ripple to the power supply of the digital circuits, which can cause logic to not function correctly.

Adding a large capacitor basically replaced the filtering and stabilization role of the original battery.

Interestingly people often intentionally remove capacitors for side channel measurements and glitching attacks.