They're trying to eliminate a step in their mandated routine. The provider (doctor, nurse practitioner, physician assistant) has to dictate their daily patient's charts so the people in the transcription department can log the events.
If they can simply record the office visit, they can eliminate that step.
There's nothing wrong with the request, because everything that goes on in the doctor's office is going down on paper either way. Either you let them record it on tape, or they're going to record it on paper. No big deal.
What do you think your patient's chart is?
If you don't trust your doctor, (which, it appears, you do not), you should find a doctor you trust.