---Edit--- So, I think you're not saying it's self inflicted, but you're wondering if that's the perception that some health staff have, and if that's why they provide poor quality treatment. If so, yes, I agree with you.
People with mental illness often find themselves at A&E. This is usualy because of their physical health - they are appropriately using A&E. (MH medication and illness can have horrible impact on physical health). But also people are often told to go to A&E if they're in crisis and not already under a MH team. That's not a great use of A&E and it's something we try to prevent, by providing better access to "crisis teams".
---end edit---
In the UK a considerable amount of money is spent on inpatient treatment for people with anorexia, and some of that is NHS funding of private for profit provision - for example Priory Group provide a lot of in-patient eating disorder treatment.
The reason it's paid for is because people who are ill deserve treatment, and a death is usually more expensive to the state than the treatment.
The reason so much private for profit provision is funded by NHS is because the NHS has underfunded mental health treatment for many years.