1. The "homeless problem", that is, the problem of mentally deranged or violent vagrants that make public spaces less usable or unwelcoming. This is difficult to impossible to measure, both in impact and in extent.
2. The "transient homeless", that is, the down-on-their-luck or otherwise situationally homeless people. This is easy to measure because these people will attempt to secure housing and services to get back on their feet.
Fixing #2, while worthwhile in itself, does nothing to fix #1. But well-intentioned people trying to fix "homelessness" find it much easier to address #2 because there are measurable outcomes, and no messy compromises to be made about civil liberties or individual freedom vs civil order. And this relies on the fact that since the same word is used, that it carries the same connotations. People hear "reduce homelessness" and they think that this means fewer people screaming about brainwaves and starting fights on the subway, but that's a completely separate issue.
EDIT: removed discussion of SROs to another top level comment to avoid confusing discourse here