From what I've heard the reason for the hatians flooding springfield is some factory wanting them for cheap labor, if we knew who the factory was maybe an email campaign or boycott could be useful.
Maybe a don't work with scummy factory campaign as opposed to a don't work with scummy magazines campaign
The answer to your hypothetical is if Dead Horse, Alaska can't draw anyone to work at McDonald's, it doesn't have a McDonald's and that is an absolutely acceptable outcome. We don't need to import an army of third-worlders to go staff a fast food restaurant somewhere that clearly isn't established enough to justify one's presence. If someone living in Dead Horse really wants to live near a McDonald's, they are the ones who need to exercise free movement.
If no American wants to go there to work at a McDonald's there and lack of garbage food isn't enough to make the current residents leave, then you have an equilibrium. The market is working just fine. Maybe some people are sad they can't have a Big Mac, but that is fine.
Hence my point. Even a completely lax and open border in a purely free market doesn't equate to hordes of people flooding in. You kill the subsidy system and the whole migration racket goes with it.