Historically visas could and were denied for completely arbitrary reasons.
The right to free speech. Even in its restrictive First Amendment form.
Once they're in the country, sure.
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Not that I think it's been demonstrated that this policy will improve US security or etc. Wonder if the APA applies here.
Textually, it does. (The President acts without force of law when he restricts “freedom of expression.”)
Intent-wise, for those seeking entry to America, it does—our republic was formed, in part, to restrict the executive from excluding religious minorities he doesn’t liked
Legally, however, you are right.
Is this true? My understanding is that the 1A has been understood by SCOTUS as a restriction on government power to influence speech, _not_ as a right granted to individuals.