I had issues with my dick so had had to drop my trousers a few times but never get naked. In fact I was surprised how for how short a time I had to drop the trousers, we'd discuss the problem, the doctor would take a quick look and then, with trousers on, discuss the treatment. On a couple of occasions I just got it out as it was easier to discuss things with my cock there and not hidden away - even if you are shy about having your dick examined once its done, its done whether for 2 minutes or 10.
After a while I was referred to a dermatolgist, a junior doctor examined my then said she would go for the consultant and he liked the patients totally naked. This was the one doctor i had who I found unpleasant and it was strange lying there naked, hand covering my bits, talking to the consultant, the other one in the background, he actually told me in the end to take my hands off to let him look !
I had several consultations thereafter- in the end I was circumcised - I have no idea whether he expected nudity on every visit or not, though as a dermatologist he may well have - but when told to go to the couch I got naked every time. It seemed easier to make the choice to be naked than to wait to be told - and in any case when it is your dick that is being examined, you are almost there anyway !