When I combined the main PCB and the Adafruit Feather, the module will not power up when it's connected to the battery through the main PCB, but it can be powered by USB connection to the Adafruit Feather. This leads me to narrow down the problem to the main PCB.
LED DID NOT LIGHT UP WHEN CONNECTED TO BATTERY
First I checked the battery. 3.9 V, normal. Then I visually trace the power from the battery to all the components that it's connected to.
First, the ON/OFF MOSFET. Referencing the specifications sheets, the functionality of the MOSFET was checked. It's working as expected.
Then, the 3.3 V voltage regulator. Again, check its performance against the specifications sheet. This component was not producing any output, despite being told to turn it on. That means that this component is faulty and needs a replacement.
While replacing the voltage regulator, I accidentally ripped the threads by pulling the voltage regulator before the solder is fully melted.
Ripped threads are difficult to fix and require someone with sufficient experience and steady hands. I found a friend who's more experienced in electronics, and he soldered a jumper wire connection with some thin magnet coil wires (orange wires in picture).
After replacing the voltage regulator and fixing the ripped thread, the combined module with Adafruit Feather is still not turning on. I consulted the schematic of the main module PCB (provided by Ali) to further troubleshoot.
After following the power source to Adafruit Feather (turns out the Adafruit Feather is powered directly via BATT) and visually inspected using a microscope, there were traces of copper near where the battery connector on the Adafruit Feather used to be. The ground threads on the battery connector were also ripped during the removal of the battery connector, but it was hard to notice. Again, using the jumper wire strategy, the ripped thread was fixed.
After both of these fixed, the combined module now powers with battery and the LED indicator turns on when the ON button is pressed.