While I think it is theoretically possible to put together a table-specific game that is simultaneously textured enough and broad enough to actually use real-world religions and their content without it being bad....I find it very difficult to believe that people can do that with something published and not have it come across very badly. It's a simple problem of compactness. A game--be it a system, a module, an adventure path, whatever--necessarily must be smaller than the body of text and tradition it references. This means you are, of necessity, presenting only a narrow perspective on the topic.
To use one of the repeatedly-cited examples above, Arthurian myth (as far as its religious elements are concerned) is the product of a doubled re-imagining of Christianity in the Medieval Period, first by the people who lived then, second by more recent authors. It's going to be nearly impossible to not present a highly elided, simplified view of Christian theology, practice, and values, and the enormous set of cultural elements that went into making "courtly love" (which is critical to the Arthur/Guinevere/Lancelot tragedy arc) are almost certainly going to be either forgotten, unmentioned, or severely simplified. But as with any system purporting to transmit value-judgments across time and geography, the devil is (sometimes literally) in the details here, and the kinds of elision and simplification that would be required to communicate the core ideas efficiently are exactly the kinds of elision and simplification used to dismiss or deride religions in real life.