Define the metric as "people helped": then bussing them out to abandon them somewhere else isn't a solution, because the adjudicators can go "yes, you made the number go down, but you did so by decoupling the metric from what it was supposed to measure, so we're not rewarding you for it".
"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.
My spouse works in the homelessness field and the correct metric to follow is number of homeless given housing. It’s the “housing first” approach. Harder to game counting amount of people directly placed into homes - someone is paying rent and maintaining a trackable occupied space that you can verify that the client is actually utilizing - and this approach cannot be gamed by “bus them somewhere else”
What many people don’t realize is just how many normal life hurdles are significantly easier to overcome with a stable housing environment, even if the client is willing and available to work. Employment, for example, has several precursors that you need. Often you need an address. You need an ID. For that you need a birth certificate. To get the birth certificate you need to have the resources and know how to contact the correct agency. All of these things are much harder to achieve without a stable housing environment for the client.