MCP73871 doesn't start to charge the battery
Quickly looking into part 3.15 of the datasheet (please, do always link those when you expect some help regarding configuration!); are you sure the THERM pin is being used correctly?

The IC expects a 10k NTC thermistor there. Seeing a 1k resistance there would probably make the IC think the battery is being seriously overheated:


  The MCP73871 device continuously monitors battery
  temperature during a charge cycle by measuring the
  voltage between the THERM and VSS pins.
  
  An internal 50 μA current source provides the bias for most
  common 10 kOhm Negative Temperature Coefficient
  (NTC) thermistors. 


Your 1 kOhm there would only produce 50 mV or 0.05V with 50 µA passing through it.


  The MCP73871 device compares
  the voltage at the THERM pin to factory set thresholds
  of 1.24V and 0.25V, typically. Once a voltage outside
  the thresholds is detected during a charge cycle, the
  MCP73871 device immediately suspends the charge
  cycle. The charge cycle resumes when the voltage at
  the THERM pin returns to the normal range. 


Which is outside of this boundary, making the charger shutdown its charging cycle.