While I agree with you that there can be a risk of skinny people missing diagnosis because they're "healthy", I think you're overestimating how well fat people are treated in healthcare. If a patient is fat, there is no further testing done. They're told to lose weight whether healthy or not, and regardless of whether it's relevant to their concerns or not. Obesity is still used as a cutoff to deny access to surgeries that will measurably improve their health, despite there often being no increased risk of complication.

As I said, I don't disagree with your issue about skinny causing medical neglect: the way our society, including medicine, blindly follows weight as the only thing that matters (examples above for fat individuals, telling skinny people with terminal illnesses they look great for having lost weight, amputating functional organs to cause malnourishment and by extension weight loss, even to folks who are arguably healthy and in a mid to low BMI range...) Is detrimental to everyone's well being.