He's not concerned about going to the doctor to get well, but more concerned about passing his medical with a cold.
I know that, and it's a ridiculous concern. It's a temporary, common condition which in no way affects one's ability to hold a medical certificate. If his sinuses are blocked the AME may well correctly advise him to refrain from flying for a short time, but it doesn't impact the ability to obtain nor hold a medical certificate.
I've always been told to not use your AME as your family doctor.
You've been told correctly. However, if you're going to see the AME anyway, there's no valid reason why not ask him for a prescription or recommendation. You're there to get a physical examination.
If you can't get a medical because you are too sick (even though probably not the case with a cold), then he is not going to just send you back home.
Medical certificates aren't issued nor denied on the basis of being "too sick." They're denied on the basis of disqualifying conditions. If you have a disqualifying condition, then you have no business obtaining nor holding nor exercising the privileges of a medical certificate until it is corrected.
If you show up with a cold, you do not have a disqualifying condition which will prevent you from holding a medical certificate. You have a temporary condition which may prevent you from exercising the privileges thereof.
If you show up with eyesight which does not meet the parameters set forth in Part 67, then you may expect to seek a waiver or obtain disqualification. If you show up with a runny nose, it's not a big deal. You understand the difference, right?