Yes, thats the route it should be offering. (Well strictly speaking, this route is the best as it goes via traffic lights and tactile paving / lowered kerbs rather than across two busy slip roads with neither of those things, but thats a different issue.)
I certainly think access=customers and access=destination should be lower priority than general public roads (assumed access=yes or designated).
More generally, I suspect there are cases where walking across an access=customers/destination area is fine and there are cases when it is not. I don’t think there is a general way to tell the difference via tagging at present - maybe it needs a new tag. In this case, the gate and the building_passage are extra clues, plus the fact that the public highway is a very similar distance.