Vanessa Domine, University professor, teacher, author & parent.
As both a parent and professor of education, my short answer is that our precious young people would most likely not actively select those subjects in which our current democratic society requires they are well-versed (translation: They don't yet know what's good for them and for the larger world). Some examples include (from a small sample size of my own offspring): World history, global studies, argumentation and debate, economics; and (what I think are particularly salient) health, physical education, and personal finance. Therefore, states require these core and non-core subjects to be taught in public schools. One could also argue that schools rely too heavily on literacy and computing skills (teaching reading and math) and do _not _spend enough time on the non-standardized-tested subjects I listed above.
It's important to keep in mind that, historically speaking, the purposes of public schooling in the United States had little to do with individual interests of students. Rather, literacy was a curricular means for the government to unite a nation and essentially prevent another Civil War from occurring. The massive number of immigrants that arrived in the United States during the early 19th century made it particularly important for the U.S. to establish a common language (English). The idea was that if everyone learned to read English and did so through common texts (like McGuffey's Eclectic Readers, the Bible, and Webster's Dictionary), then it would unite a nation. (You can read more about the Common School movement here)
Clearly the purposes of schooling have evolved considerably over the past 200 years and it is reflected in the focus on preparing workers for a global economy ("college and career readiness" is the current mantra from the federal government). Yet literacy and computing (reading and math) are still the primary skills that are taught and tested. At the same time, our Constitution clearly delineates that public education is left up to the individual states and not the federal government. Although not everyone agrees with this, either (see "Why Doesn't the Constitution Guarantee the Right to Education?" in the Atlantic Monthly).
The challenge is that there are many brilliant young people who are gifted in areas other than reading, writing, and math, yet our public education system does not serve them well because school curriculum is limited to these areas. There is a movement to educate the "whole child" that has gained a foothold (particularly in impoverished communities) that gives hope in this area (see "What Does it Mean to Educate the Whole Child?" from ASCD)