To answer your question as to what I mean when I say appropriate, the concept of FAPE, (Free Appropriate Public Education) is grounded in the landmark legislation called IDEA, Individuals with Disabilities Educational Act. It has evolved since then to become a technical concept within education that applies to all students. What is means, broadly, is that the education a student recieves is (1) designed to meet the students' unique educational needs, (2) addresses both academic needs and functional needs, (3) provides access to the general curriculum to meet the challenging expectations established for all children and (4) is reasonably calculated to enable the child to receive educational benefits.
Coming back to what you said about your belief that free education is not a right, all I can say with that statement is that it's a position that most free-market countries have moved away from. It was the de facto practice for much of American history, to treat an education as a commodity rather than a right, and it is not a modality that enjoys a great deal of support these days. You and I will simply disagree on this, because I am on the other end of the spectrum. I believe that a free appropriate public education is a right.
I will mention again that funding and governing public schools in the U.S. is handled from the bottom up, with the district enjoying the greatest power, followed by the state and federal government in decreasing levels of influence. It is not state income or sales tax that directly funds schools, but local property taxes.
Those countries I was talking about earlier that routinely do better than U.S. students are precisely those that have a greater deal of state-level control over their schools. Germany, Denmark, Sweden, Japan, etc., all have educational policy directed on the national level, with a unified state curriculum and professional standards.
And your charge that private schools provide better per-dollar value for students misses a few key pieces of information. First of all (and I'm going to have to speak in generalities because there is a wide range of variation among private schools, just as there is among public schools) most private schools do not provide transportation to and from school, which is a not inconsiderable operational expense. Most private schools do not provide food for students, although some cater to outside companies. Food service has never been a money-making strategy for public schools, even before you consider the added expense of children that qualify for free or reduced price meals. Thirdly, athletic facilities at private schools, while more prevalent than at public schools, are not universal either, whereas they are at public schools.
But finally, the single greatest reason why public schools spend more on a per-student basis is the fact of special education students. Nationwide averages indicate that 13.7 percent of all schoolchildren receive at least some special education services as part of their schooling, and the individual needs of children in these programs entails a higher cost. Educating a student with special needs costs $9000 more in a given school year than a mainstream student, and factoring that as a consideration would close the cap between private and public schoole expenditure per child vastly.