The gap between intuition and science is the math. To be "science", you need some kind of formal model that takes a state as input and returns a prediction about that state in the future.
Science is most interesting when it gives you something non-intuitive. Intuition says "all things come to a rest", which was the best science did for a few thousand years. Eventually somebody ditched intuition and said, "Actually, objects in motion tend to remain in motion. You're just confused because friction is everywhere."
You can try to re-train your intuition to match, but a lot of science toys are fun because you never really achieve that. Think about the Newton's Cradle toy, with the clacking hanging balls. The math is simple, but it's fun because it's nonetheless surprising.
If you work with the formulas long enough, you can see the patterns and re-train your expectations. Weird games like the Quantum Eraser are old hat to physicists, who are not surprised by the results -- but only because they've worked with the math enough to see where the formulas go. Their intuition never really trains itself to expect that in daily life, because daily life never works like that.
They can go from the formulas to expectations of experiments, but not the other way around, because the experiments are so unlike daily life. There's nothing to compare it to. It's hard enough with slightly unexpected phenomena like the Newton's Cradle; it's impossible with things that are truly outside of anybody's experience ever.