The design goal was to reduce the amount of current needed through the use of the huge capacitor.
Usually, the duration of 2A peak is in the us range but it is enough to cause a reset event. The capacitor tries to hold the voltage when this happens provided that the input current is not too low. I think the recommended 700-1000 mA should be enough.
When selecting battery, take a look at the maximum pulse current capability.
For example, I used a wrongly battery of Li Thionyl with 250 mA pulse current only.
The board didn't work and resets all the time. Then, I swapped to one rated for 2000 mA, then it work.