It doesn't need to make a dent. It just needs to put food in an empty stomach. After your emergency has taken every cent you have saved (in the month savings scenario) then a payment that same day will feed you.
Sure, there just aren't very many emergencies (other than someone literally stealing your money) that actually work that way. Most people are far more likely to get themselves into some kind of emergency getting paid every day than they are to get themselves out of one.