Hmm, gee. I wonder how this nation ever got started; after all, we were all immigrants without public schools. Your reasons are so weak.
In America's past the vast majority could not read or write much - nor did that matter in searching for work - being that you were a farmer or worked in the factory or other such manual labor. President Andrew Jackson was barely literate.
The common school movement as public schools were called had to fight against some (not all) industrialist forces (for whom universal education meant less availability of child labor and more taxes), with parochial schools (who perceived public education as a threat to their religious values), and with parents (children were not available to work on the farm or in factories for the family).
The late 19th and early 20th century saw the creation of a massive, secular system of public education open to all.
The movement for common schools constituted a rebuilding of the educational system (from the one school house model of old) to respond to the needs of expanding industries and the realities of growing cities such as training and disciplining of the new immigrants so they could become productive workers, and the provision of stability and control in the populated urban areas as well as educating away dangerous ideas from European immigrants such as anarchy and Communisim, etc. The common school movement was supported by the need of far seeing inustrialists/factory owners for a disciplined and productive workforce, the need of the emerging liberal state for social and ideological control (removing it from the church - lots of Catholics made our leaders fear this back then), the hopes of working class and immigrant groups for upward mobility, and the good faith and drive of many well-intentioned educational reformers who conceived education as the main avenue to build a more democratic, more egalitarian, and better society.
That is the history.
So why did the Public School system fail us in the last 30 years? I don't know.
One answer? No more factory jobs. Therefore industrialists no longer lobbied the Fed govt for such public education programs? Still reading studies on the matter myself.