It seems that the bandgap idea is fairly sensitive to getting the reference voltage right. I suppose that makes sense. If you were going to deploy this in the field you probably need to calibrate each unit (perhaps keep the figure in EEPROM).
Yes EEPROM could hold the band-gap constant value for a given specific chip, it's not like those two thing could become separated. 
I think it kind of depends on the application. If your just interested in a low Vcc alarming function then it's not really all that important that the Vcc value is 'accurate' or not, as long as it's repeatable, linear, and sensitive enough for the function, which it is, as the error is just with the constant value being used, it's not a 'variable error' with any more measurement uncertainty then if the constant value was right on value. So you could just use 1100 millivolts as the constant and just compensate for the 'error' in the setpoint value used to use to test against the reading you get. That is, is the result greater or less then the alarm setpoint value. Also as a low battery alarm a lot depends on the battery chemistry used. Lead acid and li-po have a reasonably linear discharge voltage slope, where as nicads an nimh have a very flat discharge voltage until very near the end of the charge capacity. However if the application is dealing with using the analog input pins for sensor inputs, in the presence of a gradually falling Vcc voltage, then proper calibration is probably desirable and useful.
The main thing is that this demonstrates that there is a method to maintain reasonable accuracy and calibration for reading of the analog input pin even with a variable Vcc/Avcc voltage without requiring any external components.