"Number of homeless given housing" is only the correct measure due to the nature of the domain-specific problem. I'm wary of this strategy in general, because the people responsible for deciding how things are accounted for are
rarely experts enough to identify sensible domain-specific metrics, so they'll have to consult experts. But that creates a vulnerable point of
significant interest to would-be grifters, and if they're not experts enough to assess expert consensus, you end up with metrics that don't work,
baked in.
But yes, if we're only looking at homelessness, "how many formerly-homeless people have been given housing?" is a very good way to measure successful interventions.