Both; suburban and rural homelessness is less visible, but also less common, in part because urban jurisdictions are inherently more amenable to service delivery (whether by the government or by charities) to the homeless, and homeless people are not nailed down in one place.
> Are they shifting their homeless burden to places like cities with more programs and support for homelessness?
Some of it is “shifting”, some of which is just inherent features of the places conducive to Republican political success in the first place.
> Because I've never actually heard a Republican plan for dealing with homelessness that doesn't involve criminalizing it and sweeping it under the rug where no one can see it.
To be fair, Democratic-run jurisdictions do this a lot, too (there are some exceptions, but they are exceptions even among urban Democratic jurisdictions.) The difference is that heterogenous urban jurisdictions can sweep the homeless out of the places where the people who matter can see them without sweeping them out of the jurisdiction or the range of local government or charity service delivery.
Small town police departments are notorious for running "undesirables" out of their town...redlining and zoning keeping certain people from buying real estate...things like that.
Not really. Donald Trump is, for quite a long time, much more of a personal brand marketer than a real estate developer (though “real estate developer” is part of the personal brand he markets.)