I think my pediatrician did a *great* job of warning me without making me feel like DS7 was a freak or something, as can happen in that sort of situation if the doctor isn't pretty tactful.
After DS7, then 15 months, pointed out his cranium and his ulna (since if they can learn arm and leg, they can learn the medical names, too, right?), the pediatrician said, "You know he's more than a little bright, right?" It opened the door to a conversation about giftedness and meeting DS7's needs in a positive way. I think the doctor was probably a former GT kid himself, and having met his kids, I suspect he knows GT kids from personal experience. Also, another ped in the practice had two kids who graduated from high school early, so they're all pretty comfortable with GTness. It's made everything easier!
Interestingly, at that same 15-month visit, the doctor was the first person to gently suggest that we might consider homeschooling DS7. I know I looked at him like he had three heads at the time! LOL! I reminded him of the conversation when we saw him this year, and thanked him for being supportive. He was just glad that everything was working out.
I'm not sure all pediatricians are as comfortable with and positive about GTness (let alone homeschooling!) as mine is. But if they were, I'd support your idea, kimck. Pediatricians probably interact with more different kids on a daily basis than just about anyone else. (Even teachers only see the same 25 kids every day for a year!) If any non-GT-expert might have the breadth of experience to spot a potentially GT kid in minutes, it's probably a ped!