yes,with low voltage it wont even click.If the voltage is close to the treshold it may click for jst a fraction of a second,but the power to the starter will be cut immediately to prevent the electronics from a low voltage.
You cannot test the EWS with multimeter.You need a scanner(GT1 or something else)to read the status of the EWS and ECU.
Leave the battery disconnected for 24 hrs,you can also short the terminals on the car while the battery is disconnected.Charge the battery good.