If my wife has a doctor's appointment and I need to take her to it, do I have to take a chargeable leave day?

Leave or pass, those are the only two "official" and legitimate ways to be absent from your place of duty.

That said, Using the hands and feet of every senior NCO on this forum I doubt I could count the number of times I gave "permission" for a SM to go to a family member appointment during duty hours. The only time I made it official as in pass, was if they had to leave the local area, and the only time I had to make them take leave was if it would going to take more then a normal duty day or leaving the "Norm" area in milage for a pass

The dilemma here is how is the office covered while you are gone?

From what you described I don't see much of a problem.

One of the very view direct orders I ever gave was to tell an airman to take the day off. He kept working at busy work until I told him if he didn't leave he and I were spend all weekend at work.

It Seems in one form or another if Your scheduled for duty and You can't arrange something on Your own time then You need to be excused from duty. I don't feel You should have to take leave for that but You do need approval if its not on Your own time esp if in a situation someone else has to pick up Your duties while You are at Her Appointment.