Dear Alice, like you, over the years I’ve read many books and articles about Japanese customs and culture. I’m not 100% sure but I think these trays were mentioned in Basil Hall Chamberlains “Japanese Things”. In the days of outhouses, human waste was collected in urban areas and transported to farms for fertilizer. These “honey wagons” were said to be a common sight in Tokyo. This material was much sought after by farmers and collectors paid residents for the privilege of removing it. The trays were created as a way of preventing hand to hand contact. Then, in addition to being sanitary, the trays were found to be useful and they were widely used from then on.
Alice Gordenker アリス・ゴーデンカー 
Marvellous! Really interesting question! And even better explanatory story!
Dear Alice, like you, over the years I’ve read many books and articles about Japanese customs and culture. I’m not 100% sure but I think these trays were mentioned in Basil Hall Chamberlains “Japanese Things”. In the days of outhouses, human waste was collected in urban areas and transported to farms for fertilizer. These “honey wagons” were said to be a common sight in Tokyo. This material was much sought after by farmers and collectors paid residents for the privilege of removing it. The trays were created as a way of preventing hand to hand contact. Then, in addition to being sanitary, the trays were found to be useful and they were widely used from then on.