Well, I think it does explain. The unsatisifactory nature is that the demonstration and proof are beyond our technological means.
No. It's not a technological question, but a philosophical one.
At the moment I view this in similar categories as interstellar travel or a future collider that could simulate the first trillionth of a second of the big bang: They are things that I can conceive of as extrapolations of our current technology, but which are so far removed from current technology that there's a good chance they may reveal failures in our understanding along the way.
I think that's entirely the wrong end of the stick. We actually have a fair idea, for example, of how to build such a large collider, and we have a reasonable guess as to what we could find when we built it. We can't build it, because we don't have the budget, and we don't have the materials. But we also know what kind of budget we would need in order to build it, and we know what kind of materials we would need. Any decent engineer could cost the project out, even if it ends up with an astronomical number of zeros.
It's also, more interestingly, in a different category from FTL travel which I can't conceive of as an extension of current technology or understanding.
That's a better analogy for "consciousness." There's no current technology or understanding that can possibly be extended to produce it; we simply don't know enough about it.
Immaterial, eternal, timeless? How do we know that?
Part of the traditional definitions, going back to Aristotle and before.
Broadly, in the same way we know that unicorns have one horn, are attracted to virgins, and can cure poison with their horns. Because that's how the concept is defined, whether unicorns actually exist or not. If I dug some wierd two-horned beast out of the depths of the Black Forest and claimed it was a unicorn, you wouldn't accept my claim, even if it were a new beast you'd never seen before.
I would think we know the soul must be material. It's alleged properties include the ability to cause things to happen in the physical world.
But the same list of alleged properties (see? You do know how we know the properties of the soul!) also include that it's immaterial. That's not an argument that the soul is material, it's a
reductio ad absurdam to show that the soul doesn't exist. On the other hand, since your attempted proof assumes materialism to be true (only material things can influence the material world), a non-materialist (such as a Cartesian dualist) will not find the
reductio convincing.
And what do you mean "we alone"?
Traditionally,only humans possess souls; animals do not. The question of whether animals are "conscious" is at best controversial, even among animal behavior specialists. The question of whether animals possess qualia is entirely open.
ETA: As to the advantage,... well it's the difference between saying "this drink tastes sweet" and "this drink tastes sweet because it has sugar in it." We've identify the causitive element involved. We still don't know why sugar tastes sweet (that's the qualia problem again), but we know what the element is that produces the quale.
This sort of explanation has been around a long time; objects were hot because they had phlogiston in them, for example. The concept of "heat" was reified (incorrectly) to a substance called "phlogiston." Upon further investigation, "phlogiston" turned out not to exist. But the same process led to the discovery of MSG; a Japanese chemist reasoned that there was a specific substance that tasted "delicious" and spent several years analyzing "delicious" food until he found MSG, which of course is a general purpose flavor enhancer. For reasons that we still don't fully understand.
Sugar causes sweetness. Phlogiston causes heat. MSG causes deliciousness. Soul causes consciousness. In each case, we've got a hypothesis about an underlying and unifying cause.
