The interesting question arises when one of your statements is called into question:
Once we can get a hold of a natural cause of some phenomenon, we should be satisfied that that is all there is to it.
Sometimes there's more to a phenomena than simply understanding the general forces that caused it. To give a very concrete example, I'd like to turn to cryptography. In particular, public key encryption. We can define the "natural" behavior of the RSA encryption method in terms of two keys, Key_public and Key_private. I can give you a Key_public and say "There is a Key_private such that any message encoded with Key_private can be decoded with Key_public." That's just the natural behavior of RSA. It's defined within the math. You can "get ahold of the natural cause of that phenomenon" with a small amount of math.
Now, I don't think it's extreme to say that there is a big difference between knowing that "natural cause" and actually knowing what Key_private is. In practice, knowing what Key_private is is actually quite a big deal. If you could know Verisign's private key, you could wreak quite a bit of havoc.
So this suggests that there is more to satisfaction than just knowing the causes. We also need to know the state of the universe that is being affected by these natural causes. Unfortunately, the empirical approaches humans have today cannot fully know the state of the universe. Every approach leaves unknowns in its wake, and they will continue to do so until someone discovers the meaning of everything (or disproves its existence). As an example, science runs into very interesting conundrums when their observer cannot observe systems without perturbing them.
If I may make a claim without a justification, the supernatural tends to do a better job of describing things close to one's "self." The way it handles the unobservable tends to be rather helpful for dealing with the unobservable nuances of one's own self. This is important for the rational result you are asking about: a reliance on supernatural claims would be rational if the rational entity could argue that the belief in those supernatural claims is valuable enough to warrant using non-empirical approaches to arrive at their decisions. You will find, hidden deep in that sentence, a self-referential phrasing: "the belief ... is valuable enough." Such self-referential phrasings show up quite often in places where more methodical approaches falter.
Another approach, which limits the need to explore self-referential issues, is to suggest that you don't need to know nonobservable entities exist to find it valuable to believe they exist. There are some cases where the cost of learning a law is simply too great.
Consider a case involving your only child. They are eying a thin beam over a deep crevasse. Protective instincts surge over you and you state, "Get away from that! You'll fall and hurt yourself." Did you actually mean the second sentence? It is well recognized that children will accept a statement issued with confidence more than they will accept one which accepts the possibility of being wrong, so you can maximize your value better by appearing confident. However, faking confidence is not an easy skill. It may be more valuable to actually believe your child might fall than to try to fake such a confident belief.
Likewise, consider the case where the child got half way out across the narrow beam before their confidence began to falter. You encourage them from the other side with, "Come on! You can make it!" Did you actually mean the second sentence? By the same rules as before, you can maximize your value by appearing confident, but faking it is difficult. Thus, it is in your best interests to actually believe you child can make it.
In both cases, the belief is of an unobservable. You have never seen your child fall into a crevasse. In fact, you've never seen them deal with this particular crevasse with this particular wind condition. You don't have a model of what your child will do. You cannot observe the information you need, because you only have one firstborn child (if the solution is to sacrifice the first one to science and then have another, I dare you to suggest that solution to a parent. Get a head start :). Yet, in both cases, there is value in believing a statement about the child's future, regardless of whether it can be observed. It is even possible to show that it can be valuable to believe the child can make it, or the child will never make it, in different circumstances!