It seems like he introduces these options only to settle on cause and effect as the actual root of beliefs.

Notice how, in this statement you provided, all three aspects of Hume's three kinds of relations, are already 'there', in a certain sense: "It seems like..." (resemblance), "he introduces these options" (contiguity), "only to settle on cause and effect" (cause and effect). In effect, Hume is pointing out something significant concerning causality. What is "causality"? What is it, to be a "cause"? etc. And which of the three relations really arrives on scene first, in this series? On the usual ordering, we have an impression, and then some form of contact, and then - as you point out - the "actual root" as three times removed. But this is funny: causality is supposed to come first (a causal strike), lending itself to contact (the pool stick -> the billiard ball, another ball, etc.) and all of the associated effects. What Hume could be getting at is how to clarify, in detail, this funny aspect. He does so, not by merely reversing the order, but considering causality differently. But then again, I can only offer impressions...