Looks to me that the administration is trying to weed out students who might break the law. Subjective of course, including potentially a bit of political bias. But I'm not seeing a big downside except for the delays it might introduce, budget uncertainty for many universities and probably many students blocked when they shouldn't be. The previous open-door policy seems not to have worked, with far too many students engaging in illegal activities. So it can't be a surprise that the tiller swings the other way?
As for students staying after their courses, if they are a net contributor to the USA and not illegal then why not? Someone mentioned a lack of STEM students from the USA earlier; it sounds like there's a need for this skillset?
If the US administration doesn't want to train, for example, Chinese students, then by all means block them. It's a political decision made by a President duly voted in as per the Constitution. Don't like it? Tough; turn up and vote next time. My personal opinion on this is I don't like the rhetoric surrounding all of this, but I agree with and respect the decision and will wait to see the outcome of this policy in a few months time.
And, I hope the UK does the same thing.