Does everybody with psychotic depression go to psych hospitals?

Not always: Psychotic depression usually entails impairment in reality testing. In other words delusions - usually of guilt but sometimes persecution. It's also often accompanied by strong suicidal thinking. That's why most folks with this disorder end up in the hospital. Some larger communities have intensive outpatient and partial hospital programs that can be used in such cases to avert hospitalization.

No: Usually people are admitted to a hospital if they are a danger to themselves or others and are they are unsafe outside a hospital environment. Otherwise psychotic depression can be treated with medication or (ideally) medication + psychotherapy. If the person is very disorganized, they may benefit from the structure of a residential treatment program or "halfway" program.