Absolutely yes they would. I know someone in my extended family on the streets - his problem is addiction. Lot's of people are willing to offer a roof.
I know that's anecdata - but you asked about the extreme case.
Is it because buildings refuse to house people with addiction problems, or that the extra $1/month is better used in their mind to get the next dose of the substance?
Those conditions make the offering quite different from being able to rent your own room inexpensively. One of the facets of having your own place is that you can do whatever you want there, whether it be drugs or whatever. That might not be healthy, and it's quite understandable why family would put conditions on offering a free room, but people do value their autonomy.
I know that's anecdata - but you asked about the extreme case.