The WTS did have "Sunday schools" but discontinued them. Why?
Proclaimers Book p 245-6 16 Meetings for Worship, Instruction, and Encouragement
A book entitled The Way to Paradise, written by W. E. Van Amburgh, followed in 1924. It was adapted to "intermediate students of the Bible." For a time it was used at the meetings for younger ones. Additionally, in America, "Junior Witnesses" had their own arrangements for field service. In Switzerland a youth group formed an association called "Jehovah?s Youth," for those between 13 and 25 years of age. They had their own secretary?s office in Berne, and a special magazine, Jehovah?s Youth, was edited and printed on the Society?s presses there. These youths had their own meetings and even put on Bible dramas, as they did in the Volkshaus in Zurich for an audience of 1,500.
What was taking place, however, was that an organization was developing within the organization of Jehovah?s servants. This would not contribute to unity, and it was discontinued in 1936. In April 1938, during a visit to Australia, J. F. Rutherford, the Society?s president, found that a class for children was being held apart from the convention for adults. He immediately arranged for all the children to be brought into the main convention, which was greatly to their benefit.
In that same year, The Watchtower reviewed the entire matter of separate classes for young folks in the congregation. That study again emphasized the fact that parents are responsible to instruct their own children. (Eph. 6:4; compare Deuteronomy 4:9, 10; Jeremiah 35:6-10.) It also showed that the Bible provides no precedent for segregating young ones by means of junior classes. Instead, they were to be present with their parents to hear God?s Word. (Deut. 31:12, 13; Josh. 8:34, 35) When further explanation of study material was needed, this could be given by the parents at home. Furthermore, the articles pointed out that arrangements for such separate classes were actually detracting from the house-to-house preaching of the good news. How so? Because the teachers were staying out of the field service to prepare for these classes and to conduct them. So, all separate classes for youths were discontinued.