The RAND institute, among others, concluded that the reason US citizens have turned away from graduate degrees in the sciences is a rational market response to poor pay and working conditions in favor of fields that offer better salaries and career stability. In short, there is no shortage, just a desire to pay extremely low salaries and establish exploitive working conditions - which you can do when you control someone's right to live and work in the US when these rights are difficult to come by.
Not only do I think that there's no reason to prop up a system that suppresses salaries outside tech/finance/med (or within those fields, that happens too, especially in tech), I think there's real harm in the market distortions these schemes produce.
Arguably by making postdocs underpaid, the US as a nation is getting cheap scientific advancement, in exchange for dangling the promise of eventual American citizenship (and the American dream) in front of said postdocs.
Money is a very good proxy for demand after all. If company's can easily place people for 350+k jobs that's going to lower the pay of those 350k jobs.
PS: Effectively, your describing a planned economy where someone picks the level of expected pay for each job with more foreign workers available for overpaid jobs.
The problem with this approach, of course, is that you need a judge/executive who follows the spirit of the law so that I can't get a bunch of cheap tech workers by saying "oh, job ABC in town XYZ is its own industry/sector and not comparable to SWEs at Google".