My understanding is that the internal resistance of the battery increases as the battery ages.

You basically have a voltage divider - with some of the resistance being the load (phone electronics) and some being the internal resistance of the battery. The voltage from the battery is split across these resistances. Normally the load gets almost all of the voltage.

When the battery is old and the peak current demand comes through the battery “sees” this as a lower resistance. The larger internal resistance causes a bigger voltage drop internally in the battery and the load therefore gets a lower voltage. When this is lower than the phone needs you get a brownout/reset.