I think @Kevin already gave a good answer, but I think some points should better be explained the other way around.
As I understand it, ontological naturalism says we should believe only in the entities we can discover in science. I don't have a problem with it really, though I dislike some sorts of narrative ruptures, and would suggest pre scientific people might be able to identify some parts of the world. But nothing "spooky", yeah.
I think you're confusing something here. "Supernatural" implies that there is something outside or above (super) of nature that isn't constraint by the laws of nature or doesn't or only occasionally interacts with nature.
Now an ontological naturalist believes that there are no such (supernatural) things. Everything that is part of nature is fully and at all times part of nature. So it's about what they believe, not about what they want you to believe (although that might follow from that depending on how missionary people are about their believes).
And as a consequence of being a part of nature, it would need to interact with other objects and subjects in nature which makes it describable through these interactions. Meaning sooner or later natural sciences would need to be able to find a hint of those interactions.
So in that sense there isn't a "pre-scientific" era because any an all perception of interactions happening in the real world would count as "science" in that regard. So science wouldn't really discard old civilization's descriptions of the movement of stars or tremblings of the earth or the night coming early and only lasting for a couple of minutes or whatnot, they would just reject the interpretations of "a god did it", but would try to find an in-universe explanation that doesn't rely on external indescribable forces.
That being said you might also argue that the attribution of these natural phenomena to a person could indicate their erratic and arbitrary nature and thus the difficulty in finding a good ad hoc explanation. Though again the ontological naturalism argues that there has to be an explanation if it is real (otherwise it would be supernatural).
In terms of Kevin's statement of:
"It doesn't matter whether these phenomena exist, as long as our theories can correctly predict experimental results and observations."
Yes and no. It matters a great deal whether things exist and to a degree the ontological naturalism is at the heart of science because as explained here:
"supernatural phenomena are impractical or impossible to study scientifically, regardless of whether they exist, so let's not waste our time thinking about them."
if the ontological naturalism assumptions wouldn't hold then it would be impractical or impossible to study these things and so science would be a waste of time. What doesn't matter is whether supernatural things exist. That is as long as they stay supernatural (outside of the natural world). So as long as a god doesn't interfere with the real world in a measurable sense that god could exist or not exist and it wouldn't bother a scientist (professionally), because they couldn't measure it either way. As soon as that would change and they would be always or even occasionally interfere with reality, that would be different.
However when you're talking about the existence of natural things, their existence matters a whole lot to the scientist, it's not engineering where you just want a thing to do the thing you want it to do. Scientists actually like to understand how things work. So the best tool they have in their belt is to check whether their hypothesis of how things work yield the same result as what the real world does.
Though in terms of realness, that is a necessary requirement but not a sufficient one. So if you're describing the real territory then your description (map) should align with what you measure in the real world, but just because it does, doesn't mean there aren't also 1000 and more other explanations that would work just as good. So the ultimate arbiter for a scientist is how the real world behaves.
So what is real for science are data and experiments, theories are just a tool, a narrative to explain the existing data and to fill the gaps between the data. That some of the filled gaps turn out to be false and so that the theory needs to be updated because of that is not really a failure of the method but fully intended.
Anyway, I think that scientific anti-realists claim that science has absolutely nothing to say on what "exists" without appealing to what we can conceive of as a phenomena. All theoretical unobservable entities that are postulated in science to do not exist.
Well as long as we can't observe it, it might as well not exist. But anti-realism and realism seems to take a stance in arguing that it does or doesn't exist. While the view of science is usually just "not my business, call me when we can measure it".
With respect to quantum mechanics, this is sometimes described as the "shut up and calculate" school of thought, which has considerable overlap with the Copenhagen interpretation
The problem was that at the time science had found a mythical problem for which there was no good answer and so tons of people came out of the woodwork to produce some strange theories which were often pseudo-science, untestable or whatnot and which did more to confuse and produce fringe experts than to give any insight or understanding. So instead the theorists in quantum mechanics were encouraged to rather further the understanding of their own theory and advance that because it was at least producing results that were in agreement with the experiments. Though that isn't necessarily the be all end all of how one should do science.
Now in terms of scientific anti-realism, arguing that electrons and genes aren't real because we can't see them directly... Well again those are theories that explain the data that we can see and fill the gaps that we cannot see and we keep them as long as they do that job and if we find out that they don't work, we look for better explanations.
TL;DR the ontological realist argues that everything is physical, i.e. is interacting physically with it's environment and is thus measurable by science. The anti-realist argues that things that aren't measurable don't exist, while the realist argues that there are real things existing, while the scientist usually requires the ontological realism to be true, makes no statements about the supernatural (professionally) and treats it's theories as useful rather than real. Meaning they are held, as long as they are in agreement with what experiments confirm but rejected once they are not. Which a scientific anti-realist might call instrumentalism. So ontological realism and scientific anti-realism might not be polar opposites, as both make claims about there not being something outside of physical perception. While if you consider it without instrumentalism, they'd be kinda wrong about that.