OT can do private practice- it means you can do whatever you want with your business by definition. The realistic point to address is how you are going to be paid. Either you successfully receive payment through 3rd parties or from your clients directly. Mental health is tricky because there is lack of funding even in the public sectors. The most common place for an OT to work in mental health would be a state hospital. I think the issue is that recreational therapy, nursing, and other professionals are fulfilling the role that OT once solely had. You see less and less OT even in public mental health. If you can find a specific niche then I think any private practice is possible including mental health.