He'd choose homelessness over counseling and a roof over his head?
I've had friends bolted out of their homes as teens and somehow found ways to scrape through it and provide a life for themselves. Lives where homelessness was not desired and would not be a lifestyle choice.
I'm not sure how to suggest how to help you help him seek the path you would desire for him.
Have you reached out to NAMI for in person caregiver support?
Some personalities are part of a genetic composition. Sure, there's things as parents that could have been modeled differently or better, but sometimes MI of the child gets in the way.
Be gentle on yourself