Ultimately, the problem with "shipping people elsewhere" is that you aren't necessarily solving the root causes of homelessness (unemployment/unemployability, mental illness, undocumented status, etc.). If sending them all to prison or some penal colony (a la Australia) also involved addressing those fundamental issues, then what you say makes a a lot of sense (send them to where there are available jobs and low-income housing.

Unfortunately, the reality of the world is that such social welfare considerations simply were/are not taken care of, so carting all the homeless people away simply means sweeping the problem under the rug. There is an argument to be made that the observed resurgence of homelessness in recent years is indicative of that dust pile no longer fitting under the rug anymore.