Nope, by 1918 every state had mandatory public schooling. And a significant push for education came from industry, which needed more educated workers, and had nothing to do with the Depression. The ability for teenagers to drop out & enter the workforce went unchanged throughout the depression (and long after). Though I have heard that places with job openings often reserved them for people with families to support.

By 1900, 34 states had compulsory schooling laws; four were in the South. Thirty states with compulsory schooling laws required attendance until age 14 (or higher). As a result, by 1910, 72 percent of American children attended school. Half the nation's children attended one-room schools. By 1918, every state required students to complete elementary school [0]

[0] https://en.wikipedia.org/wiki/History_of_education_in_the_Un...