Private organizations that fund programs, such as education, must rely on donors and fundraising. Private individuals must rely on their own funds to pay for education. Do you think enough dollars would be raised through a combination of donations, fundraising, and private funds to provide quality education for all?
Of course, there is no way to know this for sure, but I highly doubt that this would be possible. The sums of money needed are simply too high.
While we do not know if there is a direct relationship between how much money is spent on students and the quality of their education, we can assume that a quality education would cost at least as much as we spend on students today. Today, the government spends about $10,600 on every student in the K-12 school system. There are something in the range of 56 million students. This means that about $600 billion would have to be spent on education. If there were 300 million people in America, this would mean that every person in the country would have to donate about $2,000 each year to provide students with a quality education. (This does not take into account the costs of higher education.)
It does not seem likely that this would be possible on an ongoing basis. Of course, some large foundations and rich people would donate much more than their share per capita. However, it is hard to imagine being able to raise even $4,000 from the average family of 4 every year.
I do not think that it is possible to fund something as vast as a K-12 educational system on private donations and fund-raising alone.