To (hopefully) answer this all at once, science's limit in this case would be the human ability to see the links, but the links still trace clearly back to some sort of entity that has (at least in part) a material existence.
Indeed, it's always occurred that way hasn't it? With people's limited scientific knowledge in the past, later being proven by other, physical influences. The sole reason behind the term "God of the gaps"
But maybe this interaction lies somewhere where physical nature grows thin to the unprovable, something that is impossible to know exactly, something impossible to test or observe. To find this area, it'd be an area that, after scientific observation, it seems to grow more and more chaotic and random the deeper you research it, and at a point this randomness will have a cause from an outer, unscientific reality, a spiritual one.
God as a "spiritual concept" would have no method to interact with our world unless spiritual is a metaphoric term.
I, too, find it absurd to think the spiritual would have a way to interact with the world. People seeing ghosts taking on form or moving things, I definitely can't see that happening.
But what I can see happening is a being, specifically God, that may be spiritual or perhaps God is even beyond spiritual nature, that links between both natures, a way that they can interact with each other but definitely indirectly (through our point of view at least).
Sort of like the spiritual interacting with the physical by influencing it, considering God would have created the physical world would that not imply that God can influence the physical world?
In a similar way it's observed that two different substance worlds have this sort of relation and can interact with each other by influence - the physical and the psychological. A depressed mind can lead to an unhealthy body, and vice versa, and often if not always, the mind's patterns or way of thinking is
always influenced by the physical.