Actually, American public schools didn’t reach their final form as we know it today.
The modern US public educational system as we know it was implemented in the mid-late 19th century and modeled on the Prussian public school system in order to fulfill rising demand in public education nationwide and to modernize it to provide education to an increasing population of students at minimal expense.
Incidentally, US research universities were also modeled after those of Prussian/German research universities in the late 19th century as underscored by how UChicago and JHU were modeled after such universities whenv they were founded.