Death Note had pretty much what would be the traditional Shinigami of Japan: An ugly demon that would more or less arbitrarily kill people. The kind of Shinigami this series has is a modern amalgam of various elements from East Asia and the West, created purely for the sake of fiction (literature, anime, etc). The scythe is the most obvious thing borrowed from the West, as well as the simple job of escorting the souls of people whose time is up for mundane reasons, instead of actually killing whomever randomly. However, it won't make a very cute manga, probably, to have a skeleton walking around as the MC, so Japan had to do something about it, like make Death a cute girl. Some series, like Bleach, went a bit more in the Yama/Enma direction in the bigger picture, which could be considered more Japanese, again, or at least more East Asian. Consequently a scythe isn't a big deal in it.