Our problem isn't insufficiently harsh punishment. Our problem is unlikely punishment. The US spends more on incarceration and less on policing and the justice system. You are less likely to get caught and punished in the US but if you get punished it's harsher. Yet studies have shown that a milder punishment that happens more swiftly and with higher certainty is much better for preventing crime.
Not sure that helps with mentally ill homeless but it's at least an improvement we could aim for