There’s frequently a lot more to homelessness than just giving someone a place to live. Many of these people are mentally ill or addicted to something and cannot function or take care of themselves. There’s additional costs above just providing a place to live - like food and clothing and healthcare.

Inmates are supposed to be provided with services like meals, showers, uniforms, and healthcare so that’s part of the reason for the discrepancy in costs. I doubt there’s much addiction care or mental health care in prison though.

These people really need a better place to get help than jail. But we don’t have socialized medicine in the US, and that’s probably a huge contributor to homelessness. Just think if you couldn’t drown in medical debt, or could walk in to any clinic and sign up for addiction care or mental health assistance how many homeless people might not have ended up homeless.

Also I’m not aware of any major US city where rent or mortgage is $10k per year. A lot of cities are buying up old motels and providing support services and temporary housing. That seems to be a good start, but it probably costs more than $10k per year per person. And without free continuing healthcare a lot of people are going to end up back on the street.