Many homeless reject shelters because they would have to give up much of their personal possessions, can't have a pet, live under tight restrictions, and deal with theft from other homeless people in the shelter.
Giving people small, secure apartments would solve many of these issues along with many emergency hospital visits and many other issues.
Many shelters in the US are run be religious organizations and discriminate against gay people, too. This is particularly a problem because LGBTQ make up a sizeable chunk of homeless youth (think: kids getting kicked out by parents).
You really can't imagine why a person kicked out by their religious parents for being gay would be apprehensive about entering a homeless shelter operated by that same religious group?
Giving people small, secure apartments would solve many of these issues along with many emergency hospital visits and many other issues.