First off, I think the only time most wear a shayne coat (especially over a hat) is when its raining (and not just drizzling). Who really cares about fashion at that point?
Second, the main alternative is wearing a plastic bag. Which is for one thing, way not cool. And secondly, it problematic without an eruv. As for a shayne coat, I don’t know if there is such an opinion that one might take it off by a shayne coat (sounds shver b/c it’d be the same with any raincoat which I don’t think anyone doesn’t allow), but for sure it is considered a beged and is no problem of hotzah (carrying) on Shabbos (being as the hatcover is one garment with the rest of the coat).