The main issue with that is that regardless of good intentions pretty much everything works with money. If the system doesn't make money then it crashes regardless of the amount of empathy (add to that, immigration is an issue that over the past 15 years has made people in both EU and US lose their damn minds). In practice obviously the US needs to train it's own workers and even give those preference over foreign ones (as any country should). But at the same time not all workers are equivalent to one another and brilliant minds can come from anywhere. Sane countries will over compete for those brilliant minds. The US not doing this and other countries doing it is an inherently losing position.
If your productivity declines under a certain limit, the system breaks down. But that is not the same statement as saying that america's braindrain against other countries is necessary for America (or other places) to function in principle (as a cold turkey, it is obviously true).
I also really don't buy the "brilliant minds" argument. It really isn't like low immigration countries such as Japan, South Korea or China are lagging so far behind in key sectors because they miss out on some super geniusses.
There is nothing wrong when one country has workers without a job and the other has jobs without workers. That is a win win. But this system of braindrain has America as its biggest driver and beneficiary. While other countries do that, they usually do not do it as much and as organized (and of course they deserve the same cricism if they do not care for the country that it drains people from).
So we gotta be clear here: The Braindrain is committed by first world nations, America first and foremost, against everybody else. And it is not done out of necessity but out of the drive for more profit, just as slavery was not done out of necessity even tho it was lucrative. It does not risk breaking down if it does not pull indian tech workers, but it risked damage to poorer and less stable nations for all the decades that it went through with this.