The empirical evidence does not support the simple claim that “deinstitutionalization caused homelessness.” It supports a more nuanced claim:

Poorly implemented deinstitutionalization contributed to homelessness among some people with severe mental illness, but it was not the primary or sufficient explanation for the large-scale rise in homelessness.

I'd put it this way:

Deinstitutionalization was a contributing cause of homelessness among a subset of people with severe mental illness, particularly where psychiatric hospital closures were not accompanied by adequate community treatment, supportive housing, income support, and social services. But the evidence does not support deinstitutionalization as the principal cause of the broader rise in homelessness.

That's quite different from saying “deinstitutionalization had nothing to do with homelessness.” It clearly did have something to do with it.

The strongest evidence instead points toward “deinstitutionalization + inadequate replacement services + housing/welfare changes” as the relevant causal story.