Prediction-based medicine would allow psychiatric hospitals to compete on actually provide good treatments. You could have experts at multiple mental hospitals give predictions for the various outcomes of their treatment before admitting a new patient and then let the hospital with the best treatment prognosis take the patient.
If you want to take a more conventional approach if you consider the person in question to be unable to consent because of mental illness you could make it an option for their spouse/​family to overrule doctors the way you allow them to make decisions for other medical decisions unless a court decided about admitting the person to a mental institution.
Prediction-based medicine would allow psychiatric hospitals to compete on actually provide good treatments. You could have experts at multiple mental hospitals give predictions for the various outcomes of their treatment before admitting a new patient and then let the hospital with the best treatment prognosis take the patient.
If you want to take a more conventional approach if you consider the person in question to be unable to consent because of mental illness you could make it an option for their spouse/​family to overrule doctors the way you allow them to make decisions for other medical decisions unless a court decided about admitting the person to a mental institution.
Sounds kinda utopian even in a not-the-least-convenient-world.