We would assume that the reason that the stop start function is not working is, as you suggest, an old battery or it could also be that you are doing more short journeys and as such the battery has not reached its upper level. This means the system will not allow the stop/start to operate as its main concern is to keep enough battery power to enable the start up when first using the car. There would be little difference if you were to switch the engine off (which is good practice if you have long periods of idling).