Simply giving individuals a third of that amount would enable them to afford reasonable rent amounts.

That makes the assumption it would be spent on rent. Given no choice and having the money be directly paid to the place that houses them would not be unlike Section 8 housing. And based on what I've seen first hand and many anecdotes from a couple friends who have houses they rent, Section 8 residents often make for very bad tenants. It would be easy to take your housing for granted if it's essentially free or heavily discounted. Not in all cases, but often enough the people at the lower rungs of society are there because they can't effectively live independently. Making housing free doesn't always change that.

Rarely do idealists that want to help people who are homeless seem to understand that basic boundaries and social standards need to be met by people to actually be independent, and if that relatively low bar can't be met, the last line of compassionate defense is paternalistic government intervention such as asylums.