I think there's plenty of evidence of that not being true. Countless examples of immigrants being given free housing in hotels, and they don't end up getting jobs and contributing to the society that's helping them. Instead they mooch off the public goodwill and trash the place.
But it still beats where they came from, and hey... it's free.
Why are you singling out immigrants? The homeless in our metro area are very much domestically produced.
But you do help illustrate a concern: homelessness is ultimately a federal issue, as some states dump theirs on other more "accepting" states.