Your last paragraph is doing a LOT of heavy lifting. TLDR: the US figures should be WAY higher if you expand the definition of homelessness like those other countries do.
More research shows the U.S. rate looks lower largely because it uses a narrow, one-night "Point In Time" measure that excludes many precarious living situations other countries intentionally count. If you harmonise definitions, the U.S. does not outperform high-safety-net countries; on unsheltered homelessness in particular, it fares worse.
In UK official usage, being legally homeless often includes people the state is actively accommodating; it is not limited to street homelessness like the US PIT figure. In Australia, their figures include couch surfing (staying temporarily with other households and those in “severely crowded” dwellings). In Germany, apart from again having a more expansive definition of homelessness, their figures also include ~130k Ukrainian refugees.
Just one example: the US figures should at least include >1.2 million students experiencing homelessness.