What really bothers me about mental health issues is, like you already touched on, the idea that it's our fault for having them and that there's some sort of easy fix. Like just let go of that hatred towards those bullies - no, never.
I also really like your broken leg analogy. I saw a post the other day on Facebook that pointed out that the brain is also an organ, and sometimes our organs don't always function properly. It's considered "normal" to have your liver or kidneys or lungs not functioning properly, with diabetes or asthma or whatever, but the second your brain doesn't quite behave like a "normal" one and you have a mental illness, it's a completely different story. It shouldn't be though. It's really unfair.