I am failing to understand the purpose of the MOSFET in this circuit from Adafruit Feather board.
The FET is connected with source and drain reversed comparing to typical high-side pMOS switch, so this is not a power switch.
The combination of body diode with schottky should work like usual battery switch-over circuit, in which case what is FET doing there other than providing body diode?
My only guess is that it is there to avoid voltage drop on the diode when battery supplies power. If source is at drain voltage (via body diode) and gate is pulled to ground (via R12) then MOSFET should be fully open.
Your guess is correct. When VBUS is not present, the gate is pulled low, and the MOSFET shorts out the body diode, connecting VBAT directly to the LDO.
When VBUS is greater than VBAT, the MOSFET is cut off and the body diode is blocking, disconnecting VBAT from the circuit.