The answer to this depends on what you mean by "advent." Public high schools first appeared at all during the 1820s in Massachusetts. However, they did not become very widespread until after the Civil War and enrollments were still low in the early 20th century.
The first free public high school was set up in Boston in 1821. By 1827, Massachusetts law required all towns of more than 500 people to have a high school. However, this sort of law did not catch on and there were only about 300 public high schools in the country in 1860. (My source for these facts is America: A Narrative History by Tindall and Shi.)
While high schools increased in number after the Civil War, they were still not common in the early 1900s. As the high school movement link below says, fewer than 10% of 18 year olds graduated from high school in 1910. It was only after this, as the US became richer and education (rather than work) became possible for more teens, that high schools really "took off."
So, the answer to this depends on whether you say the "advent" was when public high schools first came to exist or if you say it was when public high schools became the norm in American society.