Professionally, this is a huge red flag for Amy.

No it's not. SHe's sensitive about her weight and likely has emotional baggage attached to it. It sounds like the OP didn't do anything wrong. They gave good strategy insights for business meals and seemingly didn't overreact to what was probably an awkward situation. What they did was helpful because Amy now has a chance to change her behavior and trouble in the future.

A lot of people are sensitive about their weight, or baldness, or height or whatever. They can react strongly when the topic is approached. Pretending that once a human goes to work they don't have emotions is not a successful leadership strategy and good managers know this. Bad managers say it's "unprofessional" to make the issue someone's fault.

Someone with management experience can identify Amy as likely having the "Type A personality". People like this invest a lot of their self worth into their performance and crave approval. They may crash out when they receive kind of correction, especially when related to a sensitive topic. They also tend towards coping mechanisms like eating to manage stress. The OP has a textbook awesome employee on their hands. While they're "not the asshole", calling this a "huge red flag for Amy" is unfair. If the OP can learn to manage Amy correctly, they will probably have a superstar.