This gets to one of the core issues. The homelessness issue, as well as mental health issues, are all left to counties in the US. This leads to bussing the homeless and it's not even clear if the cause for the homelessness is in the area the homelessness is felt. I might become homeless in New York and get on a bus to San Francisco because the weather is nicer and I hear they are nicer to homeless there. This would be much more effectively tackled at a state or better federal level. At least handling this on a state level is a hard requirement to do what you propose and create the housing where it's cheap, not where it's costly because it's nice or there are lots of jobs there for high-earners.
All that said, the federal government in the US of course would also face all kind of pushback and obstacles in part due to the way it's set up.
There was a survey (San Fransisco Homeless Count and Survey, 2022) which says 71% of the homeless in San Fransisco were living in San Fransisco at the time they became homeless, 24% were in another California county, and only 4% were out of state. But generally it makes sense that if a single county adopted housing first at a large scale these numbers might change. Additionally, the primary cause of homelessness is the severe housing shortage and the high cost of housing. So homes for the homeless should not be implemented in only cheap areas, but the expensive areas with the highest homelessness rates. This should be combined with a large increase in general housing production in these areas to mitigate the cause of the homelessness in the first place.
Most of the homeless don't want to live somewhere cheap. They'd prefer to be homeless somewhere expensive. Unless you can force them to stay in the rural housing this plan won't work.
All that said, the federal government in the US of course would also face all kind of pushback and obstacles in part due to the way it's set up.