Frazee did not sell the Red Sox until August 1, 1923, and he made further trades with the Yankees after selling Ruth.
No, No, Nanette didn't make Broadway until 1925 but was one of the most successful shows of the era.
An article people really should read, which discusses the poor relationship between Frazee and league president, Ban Johnson, WW 1, the collapse of the Red Sox in 1919 (of which Ruth had a part) was published by
SABR on Frazee