What is the advantage of a current shunt ic versus a differential ADC across the shunt?
The advantage to the INA169 is that it can measure current on the high side of the voltage, 30 or 40 volts above your logic level supply.  You could make a more accurate measurement with the ADC directly, but you must have a voltage that is both proportional to the current and in the 5-volt range of the ADC inputs, so you will need additional circuitry to perform this task.  It is a challenge to build this type of circuit, since all of the components will have to be at the precision you are looking for.  Remember that the shunt resistor you have chosen will dissipate 3.125 W so I agree with @dim that the sense resistor will probably not remain in this accuracy range as it heats up, and even board traces will have an effect on the measurement at this low resistance.