I've worked in quite a few fairly sophisticated technical environments, have many friends working in professional positions who have discussed the issue (from engineering, to academia, to medicine), and I've yet to encounter this situation where a unicorn from another country is the only option for a skill in the entire country with a population of over 300 million.
What I often do see instead are situations where local markets seeking a skill can't or don't want to compete with the labor rates needed for to attract or contract the skills they desire. They instead have to look for more leveraged labor abroad to pass their desired cost savings onto who is willing to accept living in undesired locations, working at below market rates to escape even less desirable situations, etc.
Obviously, this is anecdotal. I'm sure the case does arise where you actually need some highly specialized labor abroad but nothing near the labor pools the H1B program allows. If there are labor supply shortages, you shouldn't need to artificially inflate the market with labor through visa programs. Raise compensation and let the markets adjust to attract talent. This sort of competition is what's forced on the labor market while businesses, yet again, avoid competition when it's against their interests.
If you truly need international labor you should absolutely be required to pay near the top of market rates for the talent is truly that rare. If businesses aren't willing to pay above market rates this talent, it must not be so critical or rare. This should prevent the case of manipulating labor markets under the guise of projected artificial scarcity.
I have nothing against immigration, let people move to this country if they so desire and seek opportunity but let them do so without being leveraged by a business holding their citizenship, freedoms, and future hope of prosperity hostage while suppressing local wages, all to make a quick buck.