I’m not even sure Wizard society as a whole see mental disorders as a health problem — they seem to act as if mental problems are a moral weakness.

It does seem that way, doesn’t it? The general attitude of the Order seems to be, “Oh, there’s nothing wrong with Sirius, he’s just being a jerk.” Dumbledore’s remarks at the end about how Sirius was “too old and too clever” to be bothered by Snape’s verbal attacks comfirm that. The implication is that if Sirius didn’t behave sensibly, it was his own damn fault. But I think expecting Sirius to behave sensibly under the circumstances was like expecting a man with two broken legs to get up and walk. It just wasn’t going to happen.

Given the way Wizarding social attitudes seem to lag behind Muggle ones, I think Sirius found himself in the same situation that a lot of shell-shocked soldiers found themselves in during the early days of WWI: people who hadn’t been through the same experience didn’t believe that all these soldiers were really sick — they were just weak, or cowardly, and trying to get out of fighting.