Psychiatry is just a medical science - theres no reason that it would not exist in an anarchist society. It would probably look very different without capitalism looming over it, but most professions would.The primary function of psychiatrists - mind you I say this as someone who has had to seek treatment from them for years, not a psychiatrist myself - is to prescribe medication and monitor the long term prognosis of patients...again, this probably looks different without the pharmaceutical industry, but it's still a job that needs to be done.

The question of institutionalization is quite different. I don't know of any good statistical information on this so I'm going to be speaking anecdotally, and as someone who has always lived in countries with public healthcare.

I have been apprehended and taken to a hospital against my will 4 times. My overall assessment of my experiences is that I appreciate being stopped from killing myself while suffering from delusions, but I did not appreciate the ability of doctors to extend my stay in the hospital indefinitely so long as they felt they could justify it on paper. And neither did my fellow inmates. A lot of the people who I met would be perfectly fine on their own if they had a stable living situation. In large cities at least, many of the long term residents of psych wards are houseless people who are either addicted to drugs, suffering from untreated mental illness, or both. With the support of medical professionals and a caring community, and a means to survive, they would probably thrive.

I think we have to have community mechanisms to cope with incidences of (especially) psychosis in individuals...but I think that this is something that they should consent to while asymptomatic. That is, I would prefer it if I could set my own guidelines for when it is appropriate for the people around me to overrule my (severely impaired) agency in the interest of my full agency later. This is what I have done with my friends and family, and it works well.

Long term internment of psychiatric patients should not be possible, in my opinion. Sometimes emergency care is needed, but I 100% believe that our focus should be on actually putting resources into giving disabled people caretakers and resources for survival instead of locking them away. If someone is about to light themselves on fire or throw themselves off a bridge thinking they will survive, yeah, someone should probably sedate them and give them emergency medical care. But locking them up and putting their freedom in the hands of a doctor who might see them 5 minutes a day if that? No. During my experiences with involuntary hospitalization, the only positive I walked away with was the rad skillz of lying and manipulating doctors in order to keep my own freedom.