Good post, although the only thing I would say is that you should remove the "(yet)" from your post as empirical methods of study can never possibly conclude about that which cannot be observed. For that we have philosophy, mathematics, and metaphysics.
Oh come man. Have some faith. (pun intended) New branches and disciplines of science are regularly being created as time goes by. And also lots and lots of theories are postulated in science not by what we can see, but by what we can't see. If I remember for a long time that was the case with dark energy and dark matter, thus the "dark" in their name. Just because it isn't directly observable doesn't mean science can't dabble around the subject and rule out other reasonable possibilities one by one.
You're speaking here to the difference between the generally inductive process of the scientific method as a whole vs. the deductive process that occurs during individual scientific experiments. Yes, you are correct that many scientific hypotheses are about unobservable phenomena, but those scientific hypotheses are always based upon the results of other empirical observations. Those observations then lead us to inductively hypothesize about what caused the events that are observed.
For example, when an apple falls from a tree and we see other things falling, we can inductively hypothesize that there is likely some unobservable law (gravity, as it turns out) responsible for these allowing these objects to 'fall'. From that hypothesis we deduce an experimental design to test the strength of our hypothesis which is either accepted or rejected.
The observation of falling objects to create the hypothesis of a law that allows objects to 'fall' was, in itself, an informal experiment. The hypothesis of this informal experiment is that if something is dropped then it will fall, and we've already tested that hypothesis informally because, on Earth, we always see things fall when dropped.
What science cannot do, however, is test a purely abstract hypothesis, i.e. a hypothesis that itself was born out of non-empirical ideas. For those hypotheses, we have the aforementioned, more abstract disciplines I spoke of earlier. God as a hypothesis is purely abstract, and while such hypotheses are beyond the scope of science, they are not beyond the scope of philosophy. From a philosophical standpoint, all you need to do to prove the existence of God is to ascribe a definition to God and logically argue whether such an entity must exist by necessity (or, conversely, that it cannot possibly exist, or perhaps even that it is not possible to conclude whether such an entity exists.
Speaking to my previous sentence, I personally like the approach taken by Christopher Langan who, instead of trying to inductively exploring the idea of God based upon evidence or trying to deduce God from a series of axioms, seeks to first remove layers of logical complexity in the Universe and see what remains when there are no more layers left to remove. As it turns out, taking this approach, the case for God becomes incredibly strong.
Here's a nice article about scientific hypotheses:
https://philosophynow.org/issues/74/Hypotheses_Forget_About_It