Question: Why does the existence of "God" have to be considered a "spiritual" question and not a "scientific" one?
Sir Isaac Newton surely had no means of knowing about—let alone verifying—the existence of quantum particles. Does that mean that quantum particles only began to exist after later scientists had “discovered” them? No. And yet, by your reasoning, Newton (and his successors up to Albert Einstein) should have forgone any effort to “discover” the quantum particles simply because they had no means to prove that they existed at that time. Your argument relies solely upon your wish to segregate the search for truth (i.e., the search for what is real) into scientific and, for lack of a better term, “spiritual” claims. What is real is real and, with the right understanding and knowledge, can be proven. Perhaps we simply do not have the scientific knowledge or technology necessary to prove (or disprove) the existence of God at this time.
Further, what is to say that the existence of God is a “random” claim? I would argue that the existence of God is logically conceivable. It would all depend on what one thinks God is. Is God—as portrayed by the Judeo-Christian belief system—logically conceivable? No. But, that does not mean that God does not exist and cannot fit into a rational/logical/scientific universe. One must merely define God in different terms than those used by most religions.
Answer: You are right that Newton was not in a position to say in detail whether atoms or sub-atomic particles existed. We, like Newton, have plenty of questions about which we are simply not in a position to decide. Are there life forms in the Andromeda galaxy? In this case, we have some reasons to think life could form on other planets and in other galaxies, based on what we know of physics, biochemistry, etc. and what we know of the properties of matter in other parts of the universe we can observe—such as the Andromeda galaxy. But we have no decisive evidence. If we are being objective, we must say we think such life possible, but we don't know if it exists or not.
However, like Newton, we also already know something. Quantum mechanics is compatible with macro-level causal action. And Andromeda beings are not here on Earth amongst us, building their own civilization. Andromeda beings are not currently exerting mind-control over all earthlings. Andromeda beings would be subject to the same general physics that we are. And so on.
So as to God: if you concede that God, the all-knowing, infinitely good, immaterial creator of all existence (the God of the monotheistic religions) is not a coherent concept, then I do wonder what it is you are chiding Objectivists for denying. There are good logical reasons to conclude that a creator of existence is an incoherent notion. There is no good reason to conclude that a super-powerful intercessor is behind earthly religions. There is no fact that gives us reasons to think there is a heaven or afterlife. If your unknown thing is not to be the creator of existence, is not to be a super-powerful intercessor, and is not to provide eternal life in a supernatural heaven, then, I am with you in saying I do not know what it might be but can't rule out a variety of trivially plausible hypotheses to do with possible alien life forms.
The point is that Newton knew there were some properties that any super-small matter must have (on Earth, in his range of experience: gravity, for instance, or the properties of light he discovered). Just so, we can infer certain generalities from our context of knowledge and the range of existence we experience. Not least of these are the axioms of awareness themselves (Existence, Identity, Consciousness, Causality, and the Primacy of Existence) which apply to all experience and all existence as such, because they summarize self-evident facts about being-qua-being and awareness-qua-awareness. These known facts, which besides the axioms also include our scientific knowledge of the universe and human nature, will still exist and need to be accounted for in any expansion of our knowledge that relates to as-yet-undiscovered facts.
In other words, notions of God, insofar as they violate the axioms or require the denial of other facts of reality we already know, are false. Insofar as they deny the need to be related to the axioms or other natural facts, they are arbitrary. And insofar as anyone promotes a "scientists' God" that is neither false nor arbitrary, it is hard to see how this hypothesized entity could meaningfully be called "God" in any sense that preserves the traditional function of that idea (which is inherently supernatural).
For a fine exposition of why Objectivists deny the existence of a God, I recommend "God, Faith, and the Supernatural," a lecture by Greg Perkins.