Then argue with those other people about the things they've said; that's a separate discussion.
The issue here is that you are thinking that because you view nose rings that way, or even that because it's the more common view, all patients will feel the same. But while there are surely patients who dislike them, there are others who feel uncomfortable around people dressed up and in ties, or who won't feel comfortable enforcing their own health autonomy with someone who seems too much like an authority figure, or who are defensive and non-trusting around doctors because they're used to being judged for their own tattoos/piercings...the point is, there are effects in both directions. Now, I'm not saying that anyone should go out and get pierced to make patients feel comfortable (again, do you think people would if the studies unexpectedly showed increased trust in the majority of patients with them?) I'm simply a proponent of allowing there to be diversity within the healthcare field, including some doctors who wear piercings/tattoos if they want to. It's not the most common preference of patients, but it's also hardly the most common choice for doctors. If we let it be without jumping to judgement and acting as if a resident with a nose ring clearly doesn't gaf about their patients, things will work out. The patients who strongly dislike it will likely find a different doctor, and those who prefer it will end up with them. Just like they would in response to any other characteristic.