.....if you pay enough the workers will come.......
Not necessarily. I regularly visit two manufacturing plants that need workers. One needs fairly high skills and the other needs medium skill. Neither can attract workers and both offer competative wages and benefits. Now, if we follow your direction, increase wages to the point the workers holding jobs switch employers the product becomes uncompetative in price. All the jobs are then lost.
There are workers available at the full employment rate of let's say 5%. But those in the 5% area are unsuitable and substandard.That means the employer must hire substandard workers. Typically this means they are druggies or don't like to come to work everyday or have children that prevent orderly attendance.
On the mobility of labor, the plants in Ohio have shut down and moved, some down south. I imagine the Ohio workers now unemployed won't move because there are no unions and the wages are lower than the union wages that closed the plants. Labor is not completely mobile.
We have wrung about all the productivity increases possible. I'll cite the Christmas tree industry again. The trees are grown on NC mountain land suitable for little except trees. The industry provide a much demanded product except it is labor intensive for short periods. It doesnot provide sustainable, everyday employment. At one time, there were enough farm folk to provide the labor. They are gone or unwilling to work for low wages. The price to consumers has risen drastically as the cost of production has risen. I believe there would be no Christmas trees if the migrants were not available. A similar tale can be told for our local tomato industry where we have Mexican Foreman who have lived here for years and are pretty well integrated and work the migrants who are temporary amd maybe illegal.
We don't know how to look at and individual and determine if legal or illegal. The process has become even more difficult by the fake documents industry. We have a law but it is unenforcable because an employer, especially a small business employer, can't tell if a worker is infact illegal.