My understanding of the chip datasheet that the thermistor is there to provide fault protection, in case of battery overtemperature, and does not otherwise enter into control of the charge cycle. Charge cycle control is driven primarily by battery voltage and die temperature.
I have the MCP73871 charger from Adafruit, and it is another example of how Adafruit goes so far and then drops the ball. The thermistor in question is a 10K SMD resistor on the board, however they did include plated-through holes for a thermistor to be attached to the battery. The charger works fine for me using solar cells as the input.
So, the question is, why is the OP's charger not going into constant-voltage mode when the batteries reach the selected setpoint of 4.1 or 4.2 V. Time for some actual measurements!