The pediatrician doesn’t rely on an appeal to authority at all. She notes that removing tonsils “has been shown” to alleviate sleep issues, but we don’t know that this is an appeal to authority and we certainly can’t assume that it’s an inappropriate one.

This is the cookie-cutter flaw of circular reasoning, where the argument assumes what it sets out to prove. The pediatrician doesn't make this mistake. She provides distinct premises to support her conclusion.

The pediatrician states that tonsil removal alleviates breathing issues during sleep, but she doesn't conclude that tonsil removal is intended to alleviate these issues. Instead, she concludes that it will alleviate all breathing problems during sleep.

The pediatrician doesn’t assume that alleviating breathing issues during sleep is the only reason to remove children’s tonsils. Instead, she assumes that swollen tonsils are the only cause of these breathing issues.

The pediatrician assumes that swollen tonsils are the only cause of the breathing problems, without considering other potential causes like allergies or asthma. If there are other causes, tonsil removal might not ensure that children don’t experience any breathing problems.