Different regions can be easily gamed by opening PO Box offices in all the regions.

Well, then you check tax return of employees -- if they don't actually reside in the same region, the last person who touched the immigration form is getting a date with a rattan stick.

Fair is an odd word here, the system should work for the greatest benefit for Americans.

I think a wage threshold and weighting would be best, but others think a controlled random mixing of labour markets is better for the overall economy.

This exists. Here are the prevailing wage cutoffs for each experience level (calculated by the BLS) they use for "software developers" in SF, for example: https://www.flcdatacenter.com/OesQuickResults.aspx?code=15-1...