What I mean to ask is: if a person assumes only that he has assumptions from which the existence of the coordinating factor can be derived,
but not that he "knows/encounters" that factor,
is his trust in the coordination really valid? It just sounds like a somewhat problematic justification, and more circular than if it were claimed that he has evidence of an encounter with it.

He doesn't claim that. I claim it about him. If he presents himself as an atheist, I tell him there are indications that he isn't. All this is explained well in the fourth lecture/notebook.

It's true that in the notebook this is presented as a claim about him,
but while a philosophical proof goes from assumptions to conclusions, a theological proof goes from conclusions to assumptions.
And therefore, when a person has a chain of assumptions about the intelligibility of the world, and by means of them comes (philosophically) to infer the existence of God,
it seems that the direction of his justification for the assumptions themselves proceeds circularly:
Why does he trust the "assumptions" about the world's intelligibility? Because I reached the conclusion that there is a coordinating factor. But why does he assume there is a coordinating factor? Because he believes the assumptions.
But if so, it comes out that the source of justification for his basic assumptions is circular. God, or "Moshe" for that matter, serves here only as an entity that would be coherent with the assumptions; he isn't claiming anything about it.
By contrast, in a "pure" theological inference, where a person discovers that he is actually a hidden believer:
If asked why do you believe the assumptions, the answer will be: because in fact you believe in a coordinating factor. But here a second-order question already sounds merely skeptical.
That is, the source of justification is not the assumptions themselves but God.
So I'm asking regarding the first possibility: can it in fact serve as a justification, even though it is circular? In practice it seems, to some extent, only to preserve coherence and no more.

Okay, here you've gotten completely confused.
He starts with the fact that he trusts the assumptions. Then he asks himself why, and answers that implicitly he believes in God and therefore trusts the assumptions. There is nothing circular here. Ask him why he believes in God, and the answer will be because of some argument (not this one).

I don't understand.
You wrote, "There is nothing circular here. Ask him why he believes in God, and the answer will be because of some argument (not this one)."
But that's not precise, because if we expand the exposing argument into a "proof from all of epistemology," then certainly a situation is created in which every argument, for example the physico-theological one, is based on those very epistemic assumptions themselves.
And as the Rabbi himself wrote about "the physico-theological argument from the reverse perspective." So here it really is a circular process—because the claim that he believes in God comes as a result of the fact that he trusts the assumptions themselves, and not the other way around.
And if so, belief in God does not provide justification for the assumptions, but only confirms their coherence at most.

By contrast, if belief in God does not come from an inference about His very existence following from "philosophical" assumptions, but from a belief based on a kind of encounter with a coordinating entity—God—then indeed the argument will not be circular. And we can say that the reason and justification for his belief in the assumptions is God.

In that sense everything is circular. Trust that there was an encounter is also a basis for belief in God. The fact that you experience an encounter doesn't mean there really was one.

It's true that in the broadest sense, regarding the whole system, everything is circular to some degree.
But I don't think it's the same here, because claims about a direct encounter are more like reflective claims of looking inward into consciousness rather than outward. And so they are also different from a priori assumptions about the physical world itself.
But by contrast, using arguments such as the physico-theological proof is literally the same part of the circular argument:
Why do you assume the intelligibility of the world? Because I assume there is a coordinating factor. Why? Because if the world is indeed intelligible, apparently there is an intelligence at the beginning.
After all, you yourself explicitly wrote that the proof against naturalism is connected to the physico-theological proof, only from the opposite perspective.
So if it is indeed the same argument, only in the opposite direction, then of course it is circular.

But the point is that it sounds from the book, and from here, and from another response of yours elsewhere as well, that you also see the circular explanation—such as by using the physico-theological argument—as sufficient justification for God, and through Him justification for thinking as a whole (of which it is itself a part):
"I have evidence of an encounter with Him ****or assumptions from which His existence can be derived****. But without Him there is no meaning to evidence, since I also cannot be sure of it (because of your circle). And again you return to coherence, which is another name for subjectivism."