ND:
Okay, but the claim that the Constitution itself directly gives that power to the President in particular is objectively false. Congress may have VESTED him some power some time in the 20th Century, but the only mention of migration of any kind in the Constitution gives that power to Congress. Even then, there are several Federal statutes that limit his ability to do this sort of thing willy-nilly.
If it dealt directly with international relations, then yes. However, this particular order was not given with the intention of leverage or discussion with any of the relevant other countries, so it was not done with a foreign relations goal in mind.
And so with that in mind it comes down to whether the Federal statutes gave Trump the power to issue the order. Which has a ton of legal debate. 1182(f) is the most relevant, and so in large part this comes down to whether or not the entry of these people would be actively detrimental to US interests. Which is vastly debatable, which is why the Supreme Court wanted to test this June-October.
Even if it is ruled constitutional under those circumstances (and including the increased exceptions), this can ONLY apply to new applicants, unlike how immigration authorities had initially been applying it to INCLUDE visa holders, which was absolutely illegal.
And as a side note, as someone who calls for smaller government, as I do as well, I would think you would desire a less powerful executive authority. The powers assumed to belong to the President here are ones that I strongly feel the already too-large Executive Branch shouldn't have.