The first thing to say about the Shinto funeral is how rarely it happens, and why. Japanese religious life has long been divided by occasion rather than by allegiance: the same family takes a newborn to a shrine, marries in Shinto dress, and buries its dead through a Buddhist temple. That is not confusion or lapsed belief. It is a working division of labour that both institutions accepted for centuries, and the phrase often used of it, born Shinto and buried Buddhist, is a description rather than a criticism.
The reason on the Shinto side is kegare, the impurity attached to death, birth blood and illness. Kegare is not sin. It is a condition that separates a person from the kami until it is discharged, and the whole apparatus of shrine practice, the washing of hands and mouth at the basin, the purification wand, the periods of abstention required of priests, exists to manage it. A corpse is the strongest source of it there is. Shrines therefore do not take the dead onto their grounds, and a priest who has been near a death withdraws from shrine duty for a set period.
The result is that from the medieval period until the nineteenth century almost all Japanese funerals were Buddhist, and the temple registration system of the Tokugawa period made the arrangement compulsory in practice. A tradition with no funeral of its own was not a gap anyone in it felt the need to fill.