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I think he presents a line of reasoning that also helps to formalize why finding a convex hull works.

I don't think so. In a convex hull, you only visit points on the hull of the set, not in the interior. In this problem, you visit points in the interior too (if I understand correctly).

You get to choose the line so that this circuit is possible. If you didn’t, you’d essentially be able to form a convex hull and thus not visit all the points.

No. In fact, in the end he shows a list of approaches one might think are related but turn out not to be, and computing convex hulls is one of them.