Perhaps delegating authority to issue immigrant work visas to counties in such cases. Get skilled staff from overseas. Sure, there are ways it could be done badly and go wrong, but it could be done right.
First, rural counties are not prepared to run their own immigration systems. That kind of work is centralized for several good reasons.
Second, they’d still have to compete with richer counties, who would also love to reduce their health care costs. The price overall might drop a bit, but ultimately I think more urbanized counties would tend to win out over rural counties in such a system.
Canadian medical staff, for instance, should be able to instant-transfer. Forget the small ball with licence reciprocity between states. Licence reciprocity between nations. They will come. Just let them.
You're not going to see a massive influx of doctors leaving Switzerland, Germany or the UK to come to the US because those countries are all very similar quality of life.
However, the world is bigger than handful of Western European nations which are gleefully looking down their noses at Trump-era America, and there are literally billions of intelligent, hard working, educated people across Africa, Asia, South America and Eastern Europe who would see their lives improve in every way by moving to the US.
Evidence please? :)