AFAIK it's not illegal for a doctor to examine a patient without a chaperone present (unless there've been issues and the state Board of Medicine has required it for that specific doctor). That said, I don't know of any doctors that don't request one. Heck, some request one for any examination of a patient of a different sex/gender, and some for any intimate exam of a patient regardless of sex/gender. A minority will do it for every patient every time.

As for why, we all know at least one colleague who got accused of doing something inappropriate, generally by a patient that was upset about something else. It is my hope that most of these (usually) men are innocent, but there are severe consequences either way. The most recent case I personally know about was an ultrasound tech at my hospital. He was doing a pelvic ultrasound, the patient said he made some inappropriate comments and patted her on the thigh, the local newspaper immediately published her account of events, he was fired, and he's still unemployed a year later. That said, he did have a chaperone and she vouches for him, and no legal charges were pursued.

So, yeah, as a male physician I try to avoid anything that could give even the slightest impression of impropriety, even if the law doesn't require me to. I hate that such things are necessary.