You're assuming people even have access to healthcare.

I had depression for the best part of 20 years and I'm not sure it will ever go away entirely, but in those 20 odd years, not once did I ever have access to a neurologist or a psychologist. The mental health branch of the National Health System in the UK is very hit and miss and for me, the ONLY treatment available was medication and that involved a lot of trial and error.

If a persons brainstem is already dead, then euthanasia is irrelevant. They are already dead.