Per your hunting example, it seems similar to gambling. In any particular instance, success or failure may be the result of elements outside your control, be that the subject's free will or the complex physical interactions in a roll of the dice. That's why science favors the aggregate of many repetitions over a single case.
With your friend Frank, an expert private investigator might only need your screen name and "Frank" and from there discover his true identity. However, a PI that holds to scientific standards would be unable to say that Frank doesn't exist, even if he can't find any evidence of him. However, the probability of his existence might be reduced somewhat. A hundred PI's all given the same task and all coming up negative might further reduce the probability. But science can't prove a negative hypothesis and it always deals with probabilities, not absolute truths. My point with the Zeus example is that there is no such scientific experiment, and there probably never will be.
There are just areas that reach outside of science.
And there's where we agree. That's what I mean by "incompatible," though I might phrase it more like: There are areas outside the scope of science. Those areas can't have the scientific method applied to them because the two just aren't compatible. When a hypothesis begs the question or definitions are too loose or too broad to make concrete categorizations--to identify just a few such instances--science is helpless to provide assistance.
So I don't mean a scientist can't be religious, or that you can't do a scientific study of how religious views have changed in the last fifty years. I mean that in most ways, science can't be used to verify or disprove many aspects of the world's religions. It's just not compatible, not up to that job. The two are mostly incompatible.
Full disclosure: I'm not a scientist, but I enjoy science. I'm not religious either though, not having been raised with it. Though I did grow up in a diverse neighborhood with several religions represented. By the time I was old enough to be asked "What religion are you?" I realized I didn't have an answer and couldn't find a method of picking one that didn't feel any less arbitrary than throwing darts at a list while blindfolded. I still have not selected one and likely never will.