I, too, have carried band aids in my wallet for several years. It really keeps things on an even keel when someone gets a minor owie - which accounts for the majority of "first aid" incidents. I usually keep a decent but small FAK in my bag or brief case. I will get into it about once a year, but those are fairly significant incidents. I am a bit surprised that your instructors haven't started carrying a few band aids themselves.

The most useful item in my EDC is probably a flashlight on my key chain. It came in very handy last year when I was assessing a fellow worker with a petit mal seizure, and I use it pretty often for far less serious situations.

If it isn't on your person, it is entirely too likely that it will be unavailable in a sudden emergency.