What you say is completely true, but part of the problem is that people have very different ideas of what they are talking about when they talk about homeless people.

People who tend to complain about the homeless are generally only complaining about the visibly homeless (i.e., the people living on the street). Meanwhile the people who advocate for helping the homeless are more likely to also care about the invisible homeless who tend to make up the majority of all homeless people. These are the folks who might have jobs, live in their car, have a gym membership to shower, etc. You could spend a lot of time with many of these people and have no idea they were homeless and as a result they aren't included in many people's perception of the problem of homelessness. Therefore, it's hard for people to believe that something as simple as lowering rent prices could actually solve homelessness for a majority of homeless people.