The WTS once had a separate young person's school using books designed for them. But Rutherford stopped it saying that it was the parent's job to teach the child.
My mother was born in the early 1930s. Both her parents had converted from Methodism to the Bible Students in 1925-26. She told us that her older siblings - a brother and sister - had separate Bible classes but did not know why she didn't. I cannot be sure, but Blondie, what you write may be the explanation: They had them at one stage but they were stopped under Rutherford's dictatorship.
On a related matter, I do not think they will start Sunday School type programs for children, but if they did, I bet they'd call them something totally un-Sunday School like. The Watchtower has a way of sneaking in changes and calling them something else.
I loved the comment by an earlier poster that they thought JWs already had Sunday School: The Watchtower Study could easily pass for a Sunday School lesson, given its simplistic question and answer format.