all I would need to do is respond when the alert button is pushed and decide whether to ban someone

Banning is the big gun you try to keep in your pocket yet still let people know you will use it if you are forced to.  The art is moderating with a minimum of banning while still maintaining reasonable decorum on the forum.  I have only banned a small handful of people, as I am usually very, very slow to ban while being less slow to moderate.  Hiding or editing messages and letting transgressors know where they have crossed the line via private message is often all it takes.  Besides, bad alerts mean only that one person thinks the message is bad (and they can be a result of personal clashes more than outright violations).

Also, no, you don't have to read every message though surely it is a good idea to keep tabs on what is going on.  One good idea might be to make the acquaintance of a global mod who you can give a heads up to whenever you may be pressed for time or otherwise away from the computer so that they can kind of look in for you and keep an eye out for the alerts.